The lecture shall begin

shortlyhellip

Mathematics

Review

Umpisa

na

5M sa math lecture

1MAKINIG

2MAG-BEHAVE

3MAGTANONG

4MAGSAGOT

5MAG-ENJOY

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Mathematics

Review

Umpisa

na

5M sa math lecture

1MAKINIG

2MAG-BEHAVE

3MAGTANONG

4MAGSAGOT

5MAG-ENJOY

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Umpisa

na

5M sa math lecture

1MAKINIG

2MAG-BEHAVE

3MAGTANONG

4MAGSAGOT

5MAG-ENJOY

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

5M sa math lecture

1MAKINIG

2MAG-BEHAVE

3MAGTANONG

4MAGSAGOT

5MAG-ENJOY

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Bago ang lahat

(a) 0

(b) 1

(c) 5

(d) 7

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Determine the domain and range of

(a)

(b)

(c)

(d)

QUESTION 1

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 1 Solution

The domain of y excludes values of x that will make the

denominator zero Thus the domain is

To solve for the range we first solve for x in terms of y

5

7

xyx

1 5y x x

7 5xy y x

7 5xy x y

1 7 5x y y

7 5

1

yx

y

Therefore the

range is

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

If and

find

(a)

(b)

(c)

(d)

QUESTION 2

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Recall for functions F and G

and

QUESTION 2 Solution

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 2 Alternative Solution

SUBSTITUTE a value of x and test which choice will give

the same value

Para madali let x = 0

QUESTION 2 Solution

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION

Which will give a value of 3 at x = 0

(a)

(b)

(c)

(d)

Astig lsquodi ba

QUESTION 2 Solution

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Which of the following is a linear function

(a)

(b)

(c)

(d)

QUESTION 3

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Recall that a linear function is a polynomial function

wherein the highest power of the independent

variable is 1

QUESTION 3 Solution

(a)

(b)

(c)

(d) cannot be a linear function since x and are

in the denominator

Is QUADRATIC because of

the terms 3x2

Is LINEAR so the answer is (a)

WAIT This is also linear

The answers are BOTH (a) amp (c) Weh lsquodi nga

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

What is the equation of the linear function ywhose graph passes through the point (2 4) and

has the given slope m = 57

(a)

(b)

(c)

(d)

QUESTION 4

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

We use the slope-intercept form

QUESTION 4 Solution

STRATEGY Substitute x = 2 y = 4 and m = 57 then

solve for bHence the equation of the line is

or

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 4 Alternative Solution

Check the choices Which among the choiceshellip

1 Has slope 57

2 Has a value y = 4 when x = 2

QUESTION 4 Solution

CLUE 5 angnasa unahan

ng x at 7 ang nasa

denominator

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Determine the distance from the point ( 2 9) to

the line 3x + 4y = 2

QUESTION 5

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

No choice Solution We have NO CHOICE but use the

following formula for the distance D of a point (x0 y0)

from a line with equation Ax + By + C = 0

QUESTION 5 Solution

Before doing anything rewrite 3x + 4y = 2 as

3x + 4y 2 = 0

Then substitute the values

A = 3 B = 4 C = 2 x0 = 2 and y0 = 9

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 5 Solution

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 6

If ax2 + bx + c = 0 where a b and c are real

numbers and a ne 0 which of the following

statements is true about the discriminant D

(a) If D lt 0 the two roots are real and equal

(b) If D lt 0 the two roots are imaginary and unequal

(c) If D gt 0 the two roots are real and unequal

(d) If D lt 0 the two roots are imaginary and equal

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 6 Solution

Recall the solutions or ROOTS of the quadratic equation

ax2 + bx + c = 0 where a b and c are real numbers and

a ne 0 can be solved using the QUADRATIC FORMULA

The DISCRIMINANT D of ax2 + bx + c = 0 is the value

INSIDE THE SQUARE ROOT ie

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 6 Solution

The DISCRIMINANT D determines the type or

NATURE of solutions or roots a quadratic equation

with real coefficients has

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

As an ASIDEhellipSome UPCAT-level problems that can be solved

using the discriminant

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 7

Determine the radius of the circle whose

equation is

(a) 2

(b) 3

(c) 4

(d) 5

r

y

x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a

circle with radius r and center at (h k) is

To write x2 + y2 1048576 8x + 6y = 0 in center-radius

form complete the squareThe radius is

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 8

Find the quotient of

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 8 Solution

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 9

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 10

What is x in the equation

(a) 5

(b) 3

(c) 3

(d) 2

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 11

Evaluate

(a) 32

(b) 23

(c) 3

(d) 6

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 11 Solution

By definition the LOGARITHM of a positive number x to

the base b denoted by logb x is the POWER y of b

equal to x ie

Example log3 9 = 2 since 32 = 9 Simple lsquodi ba

CHALLENGE What is the value of

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2

(b) 2 and 3

(c) 6 and 9

(d) 9 and 6

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 12 Solution

A property of logarithm is that

Shortest solutionSUBSTITUTE the choices to the

original equation

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 13

Solve for q in the equation

(a)

(b)

(c)

(d)

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 13 Solution

NOSEBLEEEED

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Naku m

atagal

pa lsquotohellip

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 14

(a) 41

(b) 38

(c) 39

(d) 37

Faye is 5 greater than twice the age of Luigi 5

years from now Faye will be twice as old as

Luigi How old is Faye 3 years ago

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 14 Solution

Let x = Luigirsquos age

2x+5 = Fayersquos age

Age nowAge 5 years from now

Luigi x x + 5

Faye 2x + 5(2x + 5) + 5 =

2x + 10

AGE PROBLEM

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 15

(a) 10

(b) 25

(c) 20

(d) 33

Paolo can finish compiling the books in library in 25

minutes Kevin can finish it in 25 minutes while

Carmela took her 50 minutes How many minutes

will it take them if they were to compile the books

altogether

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 15 Solution

Let x = no of min they can finish the job together

No of minutes

Rate per minute

Paolo 25 125

Kevin 25 125

Carmela 50 150

Together x 1x

WORK PROBLEM

EQUATION

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 16

(a) 300

(b) 370

(c) 380

(d) 390

There are 570 students in a school If the ratio of

female to male is 712 how many male students

are there

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 16 Solution

570 students in the ratio 712

MALES FEMALES

One block =

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

As an ASIDEhellip

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 17

(a) 18

(b) 19

(c) 20

(d) 21

When each side of a square lot was decreased by

3m the area of the lot was decreased by 105 sq

m What was the length of each side of the original

lot

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 17 Solution

Let x = length of the side of the square

Lengthof a side

Area

Original x x2

New x 3 (x 3)2

EQUATION

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 18

(a) 26

(b) 27

(c) 36

(d) 37

The difference of 23 of an even integer and one-

half of the next consecutive even integers is equal

to 5 What is the odd integer between these two

even integers

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 18 Solution

Let x = 1st even integer

x + 2 = 1st even integer

EQUATION The ODD

integer in

between is

the one

AFTER 36

which is 37

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 19

(a) 53

(b) 52

(c) 51

(d) 45

Find the average of all numbers from 1 to 100 that

end in 8

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 19 Solution

The average looks like this

The numerator is actually a sum of an ARITHMETIC

PROGRESSION with first term a1 = 8 and tenth term

a10 = 98 given by The average is

then 53010 = 53

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

As an ASIDEhellip

FACT The average of the first n terms of an

arithmetic progression is just actually the

AVERAGE of the FIRST AND LAST TERM

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 20

(a) 200

(b) 220

(c) 240

(d) 260

A tank is 78 filled with oil After 75 liters of oil are

drawn out the tank is still half-full How many

liters can the tank hold

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 20 Solution

78 full 12 full

75 L

drawn out

25 L

25 L

25 L

CAPACITY

= 25(8) = 200 L

25 L

25 L

25 L

25 L

25 L

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 21

(a) After 7 min

(b) After 8 min

(c) After 9 min

(d) After 10 min

Two new aquariums are being set up Each one

starts with 150 quarts of water The first fills at the

rate of 15 quarts per minute The second one fills

at the rate of 20 quarts per minute When would

the first tank contain 67 as much as the second

tank

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 21 Solution

Let x = no of minutes

EQUATION

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 22

(a) 49

(b) 64

(c) 81

(d) 100

In a classroom chairs are arranged so that each

row has the same number If Ana sits 4th from the

front and 6th from the back 7th from the left and

3rd from the right How many chairs are there

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 22 Solution

anna

FRONT

BACK

LEFT RIGHT

NO OF CHAIRS

9 X 9 = 81

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 23

A circle with radius of 5 m and a square of 10 m are

arranged so that a vertex of the square is at the

center of the circle What is the area common to

the figures

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 23 Solution

The area common to the figures is

equal to frac14 the area of the circle 10 m

10 m

5 m

5 m

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 24

How many liters of 20 chemical solution must be

mixed with 30 liters of 60 solution to get a 50

mixture

(a) 5 L

(b) 10 L

(c) 15 L

(d) 20 L

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 24 Solution

Let x = no of L of 20 chemical solrsquon

Vol (L)

concen-tration

Amount of chemical

Sol 1 x 20 02x

Sol 2 30 60 30(06) = 18

mixture (x + 30) 50 05(x + 30)

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 24 Solution

EQUATION

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

ANG TSALAP-TSALAP

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 25

A URent-A-Car rents an intermediate-size car at a

daily rate of 34950 Php plus 100 Php per km a

business person is not to exceed a daily rental

budget of 80000 Php What mileage will allow the

business person to stay within the budget

(a) 300

(b) 350

(c) 400

(d) 450

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 25 Solution

Let x = mileage

EQUATION

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Rules of Counting

The Fundamental Principle of Counting

If an operation can be performed in n1 ways and for each of these a second operation can be performed in n2 waysthen the two operations can be performed in n1n2 ways

Extension The Multiplication Rule

If an operation can be performed in n1 ways and

for each of these a second operation can be

performed in n2 ways a third operation in n3

wayshellip and a kth operation in nk ways then the k

operations can be performed in n1n2n3hellipnk ways

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

PERMUTATION ndash based on arrangement of

objects with order being considered

Permutation of n objects

n(n ndash 1)(n ndash 2)hellip (3)(2)(1) = n (n factorial)

Permutations

Permutation of n objects taken r at a time

rn

nPrn

Permutation of n objects with repetition

1 2

k

n

n n n

Rules of Counting

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

Combination ndash based on arrangement of objects

without considering order

Combination of n objects taken r at a time

rnr

n

r

nCrn

Combinations Rules of Counting

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

13983816possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 26

How many 3-digit numbers can be formed from the

digits 1 2 3 4 5 and 6 if each digit can be used

only once

(a) 100

(b) 110

(c) 120

(d) 130

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 26 Solution

1st digit

6 choices

62nd digit

5 choices

53rd digit

4 choices

4

By the Multiplication Rule

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 27

The basketball girls are having competition for

inter-colleges There are 15 players but the coaches

can choose only five How many ways can five

players be chosen from the 15 that are present

(a) 3103

(b) 2503

(c) 3000

(d) 3003

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 27 Solution

Since order is NOT important in choosing the

five players out of 15 we use the

Combination rule with n = 15 and r = 5

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 28

A coach must choose first five players from a team

of 12 players How many different ways can the

coach choose the first five

(a) 790

(b) 792

(c) 800

(d) 752

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 28 Solution

Same as no 27

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 30

What is the perimeter of the triangle defined by

the points (2 1) (4 5) and (2 5)

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 30 Solution

We can use the DISTANCE FORMULA to compute the

perimeter of a triangle in the Cartesian plane

(ie the sum of the lengths of the sides of the

triangle)

Kaya lang lsquodi ko na realize na easy lang ang case

sa problem kasi RIGHT TRIANGLE na (See the board

for the solution) p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o

respectively and angle X is 24o find the value of s

See figure below

(a) 6

(b) 12

(c) 18

(d) 20

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 32 Solution

GEOMETRY FACT

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 33

How many possible chords can you form given 20

points lying on a circle

(a) 380

(b) 190

(c) 382

(d) 191

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 33 Solution

The number of chords can be obtained using

the Combination Rule with n = 20 r = 2

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 34

Which of the following sets of numbers cannot be

the measurements of the sides of a triangle

(a) 1 2 2

(b)

(c) 3 4 5

(d) 1 2 3

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 34 Solution

Use the TRIANGLE INEQUALITY

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side

(d) 1 2 3

1 + 2 = 3 ndash should be GREATER

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 35

The figure shows a square inside a circle that is inside the

bigger square If the diagonal of the bigger square is

units what is the area of the shaded region

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

As an ASIDEhellip

The Pythagorean Theorem and Special Right

Triangles

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 35 Solution

2

Note that

bullThe side of the

larger square is 2

(special right

triangle)

bullThe side of the

square is the

diameter of the

circle so the radius

of the circle is 1

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 35 Solution

Note that

bullThe diagonal of the

smaller square is also 2

bullIf s is the side of the

smaller square then

s

2

bullThe area of the shaded

area is then

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 36

Which of the following statements is NOT true about the

figure Parallel lines a and b are intersected by line x

forming the angles 1 2 3 4 5 and 6

(a) Angles 1 and 6 are congruent

(b) Angles 1 and 5 are

supplementary with each other

(c) Angles 3 and 4 are congruent

(d) Angles 2 and 4 are

supplementary with each other

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 36 Solution

(a) Angles 1 and 6 are

congruent (alt ext)

(b) Angles 1 and 5 are

supplementary with each

other (ext)

(c) Angles 3 and 4 are

congruent (alt int)

(d) Angles 2 and 4 are NOT

supplementary with

each other ndash they are

CONGRUENT

(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 38

How many sides does a polygon have if the sum of

the measurements of the interior angles is 1980o

(a) 11

(b) 12

(c) 13

(d) 14

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 38 Solution

The sum of the interior angles of a triangle is given by

httpwwwmathopenrefcompolygoninteriorangleshtml

QUESTION 39

An ore sample containing 300 milligrams of radioactive

material was discovered It was known that the material has

a half-life of one day Find the amount of radioactive

material in the sample at the beginning of the 5th day

(a) 9375 mg

(b) 1875 mg

(c) 375 mg

(d) 75

QUESTION 39 Solution

This can be solved using a geometric progression with

first term a1 = 300 common ratio r = frac12 and n = 5 days

QUESTION 40

A survey of 60 senior students was taken and the following

results were seen 12 students applied for UST and UP only

6 students applied for ADMU only 29 students applied for

UST 2 students applied for UST and ADMU only 10 students

applied for UST ADMU and UP 33 students applied for UP

and only 1 applied for ADMU and UP only How many of the

surveyed students did not apply in any of the three

universities (UP UST ADMU)

(a) 0 (b) 8 (c) 14 (d) 20

QUESTION 40 Solution

Using Venn Diagram

UP UST

ADMU

12

6

12 ndash UP amp UST only

6 ndash ADMU only

2

2 ndash UST and ADMU only

10 10 ndash all three

11 ndash UP and ADMU only

QUESTION 40 Solution

UP UST

ADMU

12

6

33 (12 + 10 + 1) = 10

210

1

10 5

29 (12 + 10 + 2) = 5

Add all numbers in the

circles 46

Whatrsquos outside

60 46 = 14

14

BRIEF TIPS AND TRICKS

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

QUESTION 39 Solution

This can be solved using a geometric progression with

first term a1 = 300 common ratio r = frac12 and n = 5 days

QUESTION 40

A survey of 60 senior students was taken and the following

results were seen 12 students applied for UST and UP only

6 students applied for ADMU only 29 students applied for

UST 2 students applied for UST and ADMU only 10 students

applied for UST ADMU and UP 33 students applied for UP

and only 1 applied for ADMU and UP only How many of the

surveyed students did not apply in any of the three

universities (UP UST ADMU)

(a) 0 (b) 8 (c) 14 (d) 20

QUESTION 40 Solution

Using Venn Diagram

UP UST

ADMU

12

6

12 ndash UP amp UST only

6 ndash ADMU only

2

2 ndash UST and ADMU only

10 10 ndash all three

11 ndash UP and ADMU only

QUESTION 40 Solution

UP UST

ADMU

12

6

33 (12 + 10 + 1) = 10

210

1

10 5

29 (12 + 10 + 2) = 5

Add all numbers in the

circles 46

Whatrsquos outside

60 46 = 14

14

BRIEF TIPS AND TRICKS

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

QUESTION 40

A survey of 60 senior students was taken and the following

results were seen 12 students applied for UST and UP only

6 students applied for ADMU only 29 students applied for

UST 2 students applied for UST and ADMU only 10 students

applied for UST ADMU and UP 33 students applied for UP

and only 1 applied for ADMU and UP only How many of the

surveyed students did not apply in any of the three

universities (UP UST ADMU)

(a) 0 (b) 8 (c) 14 (d) 20

QUESTION 40 Solution

Using Venn Diagram

UP UST

ADMU

12

6

12 ndash UP amp UST only

6 ndash ADMU only

2

2 ndash UST and ADMU only

10 10 ndash all three

11 ndash UP and ADMU only

QUESTION 40 Solution

UP UST

ADMU

12

6

33 (12 + 10 + 1) = 10

210

1

10 5

29 (12 + 10 + 2) = 5

Add all numbers in the

circles 46

Whatrsquos outside

60 46 = 14

14

BRIEF TIPS AND TRICKS

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

QUESTION 40 Solution

Using Venn Diagram

UP UST

ADMU

12

6

12 ndash UP amp UST only

6 ndash ADMU only

2

2 ndash UST and ADMU only

10 10 ndash all three

11 ndash UP and ADMU only

QUESTION 40 Solution

UP UST

ADMU

12

6

33 (12 + 10 + 1) = 10

210

1

10 5

29 (12 + 10 + 2) = 5

Add all numbers in the

circles 46

Whatrsquos outside

60 46 = 14

14

BRIEF TIPS AND TRICKS

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

QUESTION 40 Solution

UP UST

ADMU

12

6

33 (12 + 10 + 1) = 10

210

1

10 5

29 (12 + 10 + 2) = 5

Add all numbers in the

circles 46

Whatrsquos outside

60 46 = 14

14

BRIEF TIPS AND TRICKS

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

BRIEF TIPS AND TRICKS

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

BRIEF TIPS AND TRICKS

1 READ EACH QUESTION CAREFULLY

2 Take each solution one step at a time Some

seemingly difficult questions are really just a

series of easy questions

3 Remember thy formulas and important facts

(especially in Geometry)

4 Answer the easy items first If you canrsquot solve a

problem right away SKIP it and proceed to the

next

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

BRIEF TIPS AND TRICKS

4 Try the PROCESS OF ELIMINATION A little

guessing might work

5 Employ the EASIEST way as possible (eg

substitution shortcuts tricks etc)

6 Use your scratch paper wiselyhellip

7 If you still have time CHECK your answers

ESPECIALLY your shaded ovals

8 RELAXhellip Donrsquot panic

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

PRACTICE PROBLEMS

1 If x + y = 4 and xy = 2 find the value of x2 + y2

2 If 13 of the liquid contents of a can evaporates

on the first day and 34 of the remaining

contents evaporates on the second day what is

the fractional part of the original contents

remaining at the end of the second day

3 What is the smallest three-digit number that

leaves a remainder of 1 when divided by 2 3 or

5

4 The average of 4 numbers is 12 What is the new

average if 10 is added to the numbers

5

For more info

WEBSITEhttpmathgibeyweeblyco

m

For more info

WEBSITEhttpmathgibeyweeblyco

m

FACEBOOKMathgibey on FB

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