33052124 SAT Math Practice Test 1

7/31/2019 33052124 SAT Math Practice Test 1

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SAT MATH PRACTICE TEST 1

MULTIPLE-CHOICE QUESTIONS (25 items)

1. The figure below, not drawn to scale, shows two identical semi-circles of

radius 20 cm and a right triangle.

Find the area of the shaded part . [ Use 14.3 ]

a.2828 cm b.

21056 cm c.2456 cm d.

2172 cm

2. The length of the bases of an isosceles trapezoid are 9 and 25. Find theperimeter if the measure of each of the lower base angles is 30.

a. 34216 c.3

102364

b. 66 d.3

102332


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3. The graphs of 032 byx and 163 22 yx are drawn on the sameset of axes. Find the value of b if the graph of the straight line passes

through the center of the circle.

a. 6 b. 6 c. 8 d. 8

4. Perform the indicated operation:

x

x

xx

x

4

2

2

22

a. 22

x

x b.

2x

xc.

22x

xd.

x

x2

2

5. If M and N are positive integers and .000,000,12852 NM What is the value

of NM ?

a. 20 b. 19 c. 18 d. 17

6. Vivien traveled by boat 5 miles upstream to fish in her favorite spot.

Because of the 4-mph current, it took her 20 minutes longer to get there

than to return. How fast will her boat go in still water?

a. mph342 b. mph23 c. mph42571

d. mph24115


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7. Eugeneis going to make an open-top box by cutting equal squares from the

four corners of an11 inch by 14 inch sheet of cardboard and folding up the

sides. If the area of the base is80 square inches, then

what is the length of one side of the square?

a. in11 b. in3

30520

c. in

3

49730

d. in

4

32925

8. Which isnota factor of mymmyymmyy 22 4213 ?

a. my b. my 6 c. my 7 d. my 14

9. A line has non zero slope equal to twice the X-intercept .The slope of the

line is also equal to one-half of the Y-intercept. What is the equation of the

line in slope- intercept form?

a. 18 xy b. 88 xy c. 84 xy d. 24 xy


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10. A chemist wants to make a 10% solution of fertilizer. How much water and

how much of 30% solution should he mix to get 30 Liter of a 10 % solution

of fertilizer?

a. 20 L of water and 10 L of 30% solution of fertilizerb. 15 L of water and 15 L of 30% solution of fertilizerc. 17 L of water and 13 L of 30% solution of fertilizerd. 12 L of water and 18 L of 30% solution of fertilizer

11. Given:a

ax1

and 33 1

aaV . Find the largest possible value of a

for which xV 4 .

a. 1 b.2

53 c.2

53 d.2

53

12. Simplify:2

22

2

2

)2(

)4(

22

)2(

ba

bab

aba

b

ba

baa

a. 1 b. ba 2 c. ba 2 d. b

13. A man engages in a shooting contest. Each time he hits the target, hereceives $2 and each time he misses, he pays $1. If after 20 shots the manhas lost $2, how many times did he hit the target?

a. 7 b. 15 c. 6 d. 5


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14. Each of the following statements is true. For which statement is the

converse also true?

a. If triangles X and Y are congruent, then X and Y have the same area.

b. If triangles X and Y are similar, then the corresponding sides of X and Yare proportional.

c. If quadrilateral S is a square, then S is a rectangle

d. If quadrilateral R is a rectangle, then the diagonals of R are equal

15. The ratio of the volumes of two similar cones is 8 to 27. The ratio of their

total surface areas is

a. 2 to 3 b. 4 to 9 c. 8 to 27 d. 22 to 33

16. Given that 4 nm and that 1222 nm , calculate the value of

2nm .

a. 7 b. 6 c. 8 d. 10

17. In the figure below, two 128 rectangles share a common corner and

overlap. What is the area of the shaded part?

a. 36 b. 45 c. 48 d. 54


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18. A volcanic crater has a flattened circular top of radius 100m. As a man

walks up the side of a volcanic crater from A to B, the height above sea

level is inversely proportional to the square of the distance from the center.

Given that 500 m from the center, the man is 200 m above sea level, findthe height of the volcanic crater.

a. 800 mb. 5000 mc. No possible answerd. The given information is not sufficient to solve the problem

19. Which regular polygon has three times as many diagonals as sides?

a. octagon b. nonagon c. decagon d. dodecagon


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23. Let x denote the length of a wooden stick. Find all possible lengths of the

stick if it is known that 753 x .

a. 4

3

2 x b. 4x c. 40 x d. 0x

24.Which of the following is the solution set of the inequality 84

434

x?

a. ]7,8( b.

4

19,

4

29c.

4

19,

4

29d. none of these

25. What term must be added to xyx 304

9 2 to make it a perfect square

trinomial?

a.281y b. 121 c.

2100y d. y64


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1 c 6 a 11 d 16 c 21 b

2 d 7 d 12 a 17 d 22 d

3 a 8 d 13 c 18 c 23 c

4 c 9 c 14 b 19 b 24 c

5 b 10 a 15 b 20 c 25 c

SOLUTION NEXT PAGE


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1. The figure below, not drawn to scale, shows two identical semi-circles of

radius 20 cm and a right triangle.

Find the area of the shaded part . [Use 14.3 ]

a.2828 cm b.

21056 cm c.2456 cm d.

2172 cm

Solut ion:


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To get the area of the shaded region, we get the difference between the areaof the circle with radius 20 cm and the area of an isosceles right triangle

whose leg measures 20 cm.

Area of circle = 2r

2

2014.3

22014.3

1256

Area of right triangle =2

21 legleg

=2

4040

800

Therefore the area of the shaded part is 1256 800=456.

The area of the shaded region is 456 cm2


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2. The length of the bases of an isosceles trapezoid are 9 and 25. Find theperimeter if the measure of each of the lower base angles is 30.

a. 34216 c. 3

102364

b. 66 d. 3

102332

Solut ion:

Total perimeter is 2593

3162

343

332

3

102332

Recall:

Theorem. In a 30O

60O

90O

triangle the sides are in the

ratio 3:2:1 .

To get the length of this, divide

8 by 3 , then multiply by 2.

The result is

3

3162

3

8

This is also

3

316


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3. The graphs of 032 byx and 163 22 yx are drawn on the sameset of axes. Find the value of b if the graph of the straight line passes

through the center of the circle.

a. 6 b. 6 c. 8 d. 8

Solut ion:

The center of the circle 163 22 yx is 0,3 .

We substitute the point 0,3 in the equation 032

byx

and solve for b.

032 byx

00332 b

6b


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4. Perform the indicated operation:

xx

xx

x

4

2

2

22

a. 22

x

x b.

2x

xc.

22x

xd.

x

x2

2

Solut ion:

x

x

xx

x

x

x

xx

x

42

2

242

2

222

x

x

x

x

42

2

2

42

2

2

x

x

x

x

2222

xx

x

x

x

22

x

x


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5. If M and N are positive integers and .000,000,12852 NM What is the value

of NM ?

a. 20 b. 19 c. 18 d. 17

Solut ion:

610128000,000,128

67 522

667 522

613 52

Therefore 6,13 NM

19 NM


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6. Vivien traveled by boat 5 miles upstream to fish in her favorite spot.

Because of the 4-mph current, it took her 20 minutes longer to get there

than to return. How fast will her boat go in still water?

a. mph342 b. mph23 c. mph4

2571d. mph24115

Solut ion:

Let x=speed of the boat in still water

So x+4=speed of the boat downstream [ because the speed of the current is 4 mph]

x-4=speed of the boat upstream

Use rdtr

td

3

1

downstreamupstreamtt [we convert 20 minutes to hours, 20 minutes = 1/3 hours]

3

1

4

5

4

5

xx, Multiply both sides by 3(x-4)(x+4)

16415415 2 xxx

1660156015 2 xxx

1362 x

mph342136 x


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7. Eugeneis going to make an open-top boxby cutting equal squares from the

four corners of an 11 inch by14 inch sheet of cardboard and folding up the

sides. If the area of the base is 80 square inches, then

what is the length of one side of the square?

a. in11 b. in3

30520

c. in

3

49730

d. in

4

32925

Solut ion:

Area of base = 80214211 xx

80214211 xx

407211 xx

037252 2 xx

Using quadratic formula

2237242525

2

x , we disregard the

negative value

in4

32925 x


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8. Which isnota factor of mymmyymmyy 22

4213 ?

a. my b. my 6 c. my 7 d. my 14

Solut ion:

There is a common binomial factor my , factor it out, the result is

22

4213 mymymy

The complete factored form of mymmyymmyy 22 4213 is

mymymy 76 .

my 14 isnota factor of mymmyymmyy 22 4213 .

mymy 76


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9. A line has non zero slope equal to twice the X-intercept .The slope of the

line is also equal to one-half of the Y-intercept. What is the equation of the

line in slope- intercept form?

a. 18 xy b. 88 xy c. 84 xy d. 24 xy

Solut ion:

bmxy , m=slope and b=y-intercept

x-intercept of y=mx+b ism

b , why? Because when y=0 then

m

bx

m

bm 2 slope equal to twice the X-intercept

2

bm slope equal half the Y-intercept

mb 2 , substitute in the first equation

m

mm

22

4

m

mb 2

8 b

So the answer is

84 xy


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10. A chemist wants to make a 10% solut ion of fertilizer. How much water and

how much of 30% solution should he mix to get 30 Liter of a 10 % solution

of fertilizer?

a. 20 L of water and 10 L of 30% solution of fertilizerb. 15 L of water and 15 L of 30% solution of fertilizerc. 17 L of water and 13 L of 30% solution of fertilizerd. 12 L of water and 18 L of 30% solution of fertilizer

Solution:

Let x = amount of water

30 x = amount of 30% solution

Equation: 301.0303.00 xx

33.09 x

63.0 x , divide both sides by 0.3

20 x , and 1030 x

The chemist needs 20 L of water and 10 L of 30% solution of fertilizer to get 30 Lof a 10% solution of fertilizer.

0% +30% = 10%

x 30-x 30


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11. Given:a

ax1

and 33 1

aaV . Find the largest possible value of a

for which xV 4 .

a. 1 b.2

53c.

2

53 d.

2

53

Solution:

xV 4

a

aa

a1

41

3

3

, factor the left side

aa

aa

aa

14

11

12

2

01

41

11

2

2

aa

aa

aa

04

11

12

2

aa

aa

0

1

a

aor

041

12

2

a

a

Recall:

2233 babababa


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First equation:

01

aa

012 a

12 a

1a

Second equation:

041

12

2

a

a

041 224 aaa

013 24 aa

013 24 aa

Let 2ay

0132 yy

Using quadratic formula

2

53y

2

532 a


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2

53a

The values of a are 1 ,2

53

, 2

53

The largest possible value of a is2

53

12. Simplify:2

22

2

2

)2(

)4(

22

)2(

ba

bab

aba

b

ba

baa

a. 1 b. ba 2 c. ba 2 d. b

Solution:

bababbaba

baa

b

ba

babaa

22

22

22

2)2(

1


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13. A man engages in a shooting contest. Each time he hits the target, hereceives $2 and each time he misses, he pays $1. If after 20 shots the manhas lost $2, how many times did he hit the target?

a. 7 b. 15 c. 6 d. 5

Solut ion:

Let x= number of times a man hits the target

After 20 shots the man receives/lost

2202 xx

2203 x

6x

14.Each of the following statements is true. For which statement is the

converse also true?

a. If triangles X and Y are congruent, then X and Y have the same area.

b. If triangles X and Y are similar, then the corresponding sides of X and

Y are proport ional.



Solut ion:

Recall:

The converse of the statement If p then q is If q then p.


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6

8

1012

4

104

a. If triangles X and Y are congruent, then X and Yhave the same area.

The converse of statement a is

If triangles X and Y have the same area then X and Y are congruent .

Counterexample:

Area= 24 Area= 24 Two triangles have the same area but are not congruent. Therefore the

converse of statement a is false.

b. If triangles X and Y are similar, then the corresponding sides of X andY are proport ional.

The converse of statement b is

If the corresponding sidesof triangles X and Y are proportional then X

and Y are similar.

Which is also true

Answer is b.


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We will show that the converse of c and d are false.


The converse of statement c is

If quadrilateral S is a rectangle then S is a square.

Counterexample:

The figure above is a rectangle, but it is not a square. Therefore the

converse of statement C is false.


The converse of statement d is

If the diagonals of quadrilateral R are congruent, then r is a rectangle

Counterexample:


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The diagonals of the quadrilateral (isosceles trapezoid) above are

congruent, but it is not a rectangle. Therefore the converse of

statement d is false.

15.The ratio of the volumes of two similar cones is 8 to 27. The ratio of their

total surface areas is

a. 2 to 3 b. 4 to 9 c. 8 to 27 d. 22 to 33

Solut ion:

27

8

2

1 V

V, therefore the ratio of the corresponding sides [circumference

of base, height, etc] is

3

2

27

83

2

1 s

s, and the ratio of their total surface areas is

94

32

2

2

1

A

A


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16. Given that 4 nm and that 1222 nm , calculate the value of

2nm .

a. 7 b. 6 c. 8 d. 10

Solut ion:

4 nm , get the square of both sides

162 22 nmnm

16222 mnnm

16212 mn

42 mn

Therefore

222 2 nmnmnm

mnnm 222

412

8


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17.In the figure below, two 128 rectangles share a common corner and

overlap. What is the area of the shaded part?

a. 36 b. 45 c. 48 d. 54

Solut ion:


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Area of shaded part = Area of 8x12 rectangle Area of the two right triangles

formed by connecting one corner of the rectangle to the intersection point

along the right edge of that rectangle

2

1122

98128

63696

54

18. A volcanic crater has a flattened circular top of radius 100m. As a man

walks up the side of a volcanic crater from A to B, the height above sea

level is inversely proportional to the square of the distance from thecenter. Given that 500 m from the center, the man is 200 m above sea

level, find the height of the volcanic crater.

a. 800 mb. 5000 mc. No possible answerd. The given information is not sufficient to solve the problem


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Solut ion:

Let levelseaaboveheighth and centerthefromdistanced

2dkh , where k is a constant of variation

When 500d , 200h

2500200

k 7105 k

To get the height of the volcanic crater we let 100d , which is the radiusof the top radius and solve for h .

So 2

7

100

105h

mh 5000


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19. Which regular polygon has three times as many diagonals as sides?

a. octagon b. nonagon c. decagon d. dodecagon

Recall:

If a polygon has n sides then the number of diagonals is equal to

2

3nn

nnn

32

3

nnn 632

092 nn

09

nn

9n disregard 0n

Answer is nonagon


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20. Find the equation of the circle with center in the first quadrant and is

tangent to the lines 7y , 9y , and 4x .

a. 6441 22 yx

b. 161422

yxc. 6414 22 yx

d. 10022 22 yx

Solut ion:

First we graph the tangent lines.

Based on the graph of the tangent lines , the y-coordinate of the center

of the circle is

12

97

,

So the radius is equal to 87119

Recall:

The equation of the

circle with center at

kh, and radius ris

222 rkyhx


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The x-coordinate of the center is 8 units to te right

of 4x or 4.

Therefore the center of the circle is 1,4 and the equation is 6414 22 yx

The graph of the circle is


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21. Simplify:

2

1

2

1

2

4

1

2

1

23

432

a. 16 b.3

32c.

3

16d.

3

8

Solut ion:

2

1

2

1

2

1

2

1

2

1

2

1

2

4

1

2

1

23

1632

23

432

2

1

2

1

2

1

2

1

33

1622

3162

3

32


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22.Which of the following statements isALWAYStrue for any real numbers

a,b,c,d?

a. If ba , then bcac

b. If ba , thenba

11

c. If ba and dc , then bdac .

d. If ba and dc , then dbca

Solut ion:

a. If ba , then bcac is only true when c is a posit ive number.

b. If ba , thenba

11

is false when a or b is zero.

c. If ba and dc , then bdac .

5 -2 -1 -7 -5 14

false


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d. If ba and dc , then dbca is always true. why? Seeproof below

ba given

cbca * add c to both sides

dc given

dbcb * * add b to both sides

dbcbca * and **

dbca

23. Let x denote the length of a wooden stick. Find all possible lengths of

the stick if it is known that 753 x .

a. 43

2 x b. 4x c. 40 x d. 0x

Solut ion:

7537753 xx

, add 5 to both sides

1232 x

, divide both sides by 3

4

3

2

x

The length of the wooden stick cannot be negative

Therefore40 x


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24.Which of the following is the solution set of the inequality 84

434

x?

a. ]7,8( b.

4

19,

4

29 c.

4

19,

4

29 d. none of these

Solut ion:

44

438

4

434

xx

and8

4

43

x

First inequality:

4

4

43

x

1643 x

194 x

4

19 x

We get the intersection of 4

19x

and 4

29x

, the result in set intervalnotation is

4

19,

4

29

Second inequality:

8

4

43

x

3243 x

294 x

4

29 x


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25.What term must be added to xyx 304

9 2 to make it a perfect square

trinomial?

a. 281y b. 121 c. 2100y d. y64

Solut ion:

We get the square root of the first term

xx2

3

4

9 2

?

2

3230

xxy

Therefore we should add 210y

or

2100yto make

xyx 304

9 2

a perfect square trinomial.

2

22 102

310030

4

9

yxyxyx

y10

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33052124 SAT Math Practice Test 1