Catholic High Emath Paper1

1

CATHOLIC HIGH SCHOOL PRELIMINARY EXAMINATIONS (3) 2008SECONDARY FOUR MATHEMATICS

Subject : Mathematics Paper 1

Level : Secondary 4 Date : 12 September 2008

Marks : 80 Time : 0815 – 1015

Name : ____________________________________ ( )

Class : Sec 4 - ____

This question paper consists of 20 printed pages, including this cover page.

INSTRUCTIONS TO CANDIDATES :

Write your NAME, CLASS and INDEX NUMBER in the spaces at the top of this page.

Answer all questions.Write your answers in the spaces provided on the question paper.If working is needed for any question, it must be shown with the answer.Omission of essential working will result in loss of marks.

You are expected to use an electronic calculator to evaluate explicit numerical expressions.If the degree of accuracy is not specified in the question, and if the answer is not exact, give your answer to 3 significant figures. Give answers in degrees to one decimal place.For π, use your calculator value or 3.142, unless the questions requires the answer in term of π.

The number of marks is given in brackets [ ] at the end of each question or part question.The total of the marks for this paper is 80.

For Examiner’s Use Only:

Units

Fractions

Brackets

Accuracy

Qn No.Types

Units

Fractions

Brackets

Accuracy

Qn No.Types Qn No.Types

Others

Geometry

Diagrams

Graphs

Qn No.Types

Others

Geometry

Diagrams

Graphs

F o r E x a m i n e r ' s U s e

8 0

2

Mathematical Formulae

Compound interest

Total Amount = Pn

r

+

1001

Mensuration

Curved surface area of a cone = rlπ

Surface area of a sphere = 4 2rπ

Volume of a cone = hr2

3

1π

Volume of a sphere = 3

3

4rπ

Area of triangle ABC = Cab sin2

1

Arc length = θr , where θ is in radians

Sector area = θ2

2

1r , where θ is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin== .

bccba 2222 −+= cos A

Statistics

Mean = ∑∑

f

fx

Standard deviation = 2

2

−

∑∑

∑∑

f

fx

f

fx

3

Answer all the questions in the spaces provided on the Question Paper.

1 Light travels 1 metre in 3.3 nanoseconds.

Find the total distance in metres, that light will travel in 6.6 microseconds.

Answer ___________________ m [1]

2 In Iceland, the highest air temperature recorded is 30.5 °C.

The lowest air temperature recorded is 7.39− °C.

Find

(a) the difference between the two temperatures.

(b) the mean of the two temperatures.

Answe

r

(a

)___________________ °C [1]

(b

)___________________ °C [1]

3 Find the integer values of x for which 2

154531

+≤+<− xxx .

Answer ______________________ [3]

4

4 23,0,5.2,,3

1,1232 −× π

(a) Complete the following table using the list of numbers provided above.

5 A polygon has n sides. Two of its exterior angles are 23º and 85º, while the other ( )2−n

exterior angles are 14º each. Calculate the value of n.

Rational Numbers:

Integers:

[2]

5

Answer ______________________ [2]

6 Solve ( ) 10063 2 =−x .

Answer ______________________ [2]

7 Factorise bxaybyax 4520 +−− .

Answer ______________________ [2]

8 Simplify ( )a

baa3203 ×

, leaving your answer in index form.

6

9 The table shows the population statistics of Singapore from 2005 to 2007.

Total population comprises Singapore residents and non-residents.

(a) Find the number of non-residents in Singapore in 2006, leaving your answer in

standard form.

(b) Calculate the percentage increase in the total population from 2005 to 2007.

Answe

r

(a

)______________________ [1]

(b

)______________________ [2]

10 (a) Express each of the numbers 66 and 168 as product of prime factors.

(b) Find the highest common factor of 66 and 168.

(c) Find the smallest integer value of n for which 66n is a multiple of 168.

Answer ______________________ [2]

YearTotal Population

(Millions)

Singapore Residents

(Millions)

2005 4.27 3.47

2006 4.40 3.53

2007 4.59 3.58

7

Answe

r

(a

)66 = _________________

168 = _________________ [2]

(b

)______________________ [1]

(c

)______________________ [1]

11 (a) The first five terms of a sequence are 1, 3, 5, 7 , 9, 11.

Find in terms of n , the n th term of the sequence.

(b) Using the answers from part (a) or otherwise, write down an expression, in terms

of n, for the n th term of the sequence

(i) 1, 9, 25, 49, 81, 121, ……….

(ii) 25, 49, 81, 121, ……….

Answe

r(a) ______________________ [1]

(b)(i

)______________________ [1]

(ii) ______________________ [1]

12 A box contains 5 red balls, 3 black balls and 1 white balls.

Two balls are taken from the bag at random, without replacement.

Find the probability

(a) that both balls are white,

(b) at least one ball is black.

A third ball is now taken from the box at random.

(c) Find the probability that none of the three balls is red.

8

Answe

r(a) ______________________ [1]

(b) ______________________ [2]

(c) ______________________ [2]

13 (a) y is inversely proportional to 3x .

9=y when 3=x .

Find y when 10=x .

(b) p is directly proportional to 2q .

q is increased by 50%.

Find the percentage increase in p .

Answe

r

(a

)______________________ [2]

(b

)______________________ [2]

14 ε = { x : x is an integer and 100 ≤≤ x }

A = { x : x is divisible by 3 }

B = { x : x is a prime number }

(a) Draw a Venn diagram to illustrate this information. Insert all elements of ε, A and

B in the Venn Diagram.

Answer (a)

[2]

9

(b) Write down ( )BAn ∩ .

(c) List the elements in the set 'BA ∪ .

Answer (b) ___________________________ [1]

(c) 'BA ∪ = ____________________ [1]

15 In the diagram, A is ( )2,6− , B is ( )2,4 and C is ( )k,12 .

y

xA ( - 6 , 2 ) B ( 4 , 2 )

C ( 1 2 , k )

0

(a) Given that A, B and C form an isosceles triangle such that AB = BC, show that the

value of k is 8.

Find

(b) the midpoint of AC.

(c) the gradient of line BC.

(d) the equation of the line which passes through the midpoint of AC and is parallel to

BC.

(e) the area of triangle ABC.

Answer (a) ______________________________________________________________

______________________________________________________________

______________________________________________________________ [1](b

)( _________ , _________ ) [1]

10

(c

)______________________ [1]

(d

)______________________ [1]

(e

)________________ units 2 [1]

16 (a) Sketch the graph of ( ) 221 +−= xy .

Answe

r

(a

)

y

x

[2]

(b)

y

xO

The sketch represents the graph of

nxy = .

Write down a possible value of n.

Answer (b) n = ________________ [1]

(c)

11

y

xO

1

Write down a possible equation for the

graph

Answer (b)_____________________

_[1]

17

h c m

r c m

C o n t a i n e r A C o n t a i n e r B

r c m

h2

c m

h2

c m

The containers shown in the diagrams has height h cm.

Their other dimensions are as shown.

The containers are being filled to the brim with water which flows into each one at the

same constant rate.

It takes 2.5 minutes for the water to reach a depth of 2

hcm in container A .

(a) Find the time taken for the water to reach the brim of

(i) container A,

(ii) container B.

12

Answe

r

(a)(i

)______________ minutes [1]

(ii) ______________ minutes [1]

(b) On the grid in the answer space, sketch the graph showing how the depth of the

water in each container varies with time.

Answer (b)

0 2 0 4 0

D e p t h o f

w a t e r ( c m )

T i m e ( m i n u t e s ) 1 0 3 0

h2

h

[2]

13

18 ABCD is a parallelogram and E is a point on AB.

BD and CE meet at X.

A B

CD

X

E

(a) Prove that triangles BEX and DCX are similar.

Answer (a) In triangles BEX and DCX, _________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________ [2]

(b) It is given that ABAE =4

Find the ratio

14

(i) area of BEX∆ : area of DCX∆ ,

(ii) area of BCX∆ : area of parallelogram ABCD .

Answe

r

(b)(i

)______________________ [1]

(ii) ______________________ [1]

19 The diagram is the speed – time graph for the first 20 seconds of a journey.

0 2 4 6 8 1 0 2 01 2 1 4 1 6 1 8

5

1 0

1 5

2 0

2 5

S p e e d ( m e t r e s p e r s e c o n d )

T i m e ( t s e c o n d s )

(a) Find

(i) the deceleration when 9=t ,

(ii) the speed when 17=t ,

(iii) the average speed for the last 10 seconds.

15

Answe

r

(a)(i

)_________________ m/s 2 [1]

(ii) __________________ m/s [1]

(iii) __________________ m/s [2]

(b) Part of the distance – time for the same journey is shown in the answer space.

Complete this graph.

Answer (b)

0 2 4 6 8 1 0 2 01 2 1 4 1 6 1 8

2 0

4 0

6 0

8 0

1 0 0D i s t a n c e T r a v e l l e d

( m e t r e s )

T i m e ( t s e c o n d s )

1 2 0

1 4 0

1 6 0

1 8 0

[2]

16

20 The points P and Q are ( )6,1 − and ( )4,7 − respectively.

The point R is such that QR =

−4

2.

(a) Find the coordinates of R.

Answe

r

(a

)______________________ [2]

(c) It is given that RS =

−h

12.

Find the two possible values of h which will make PQRS a trapezium.

You may use the grid below to help you with your investigation.

17

P

Q

Answe

r

(c

)h = _______ or _______ [2]

21 In the diagram, ABCD is a square and DEFG is a rectangle. EAB is straight line.

D C

BAE

F

G

(a) Show that angle ADE = angle CDG.

(b) Prove that triangle ADE is congruent to triangle CDG.

(c) Hence, show that DEFG is a square.

Answer (a) _______________________________________________________________

__________________________________________________________________

18

__________________________________________________________________

__________________________________________________________________ [2](b) In triangles ADE and CDG, _______________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________ [3](c) _______________________________________________________________

__________________________________________________________________

__________________________________________________________________ [1]22 The plan of a triangular field has a scale of 1 cm to 50 m .

(a) Express this scale in the form 1 : n .

Answe

r

(a

)______________________ [1]

The diagram below is part of a scale drawing of the field.

Another point, C , is 300m from B on a bearing of 063°.

(b) Complete the map to show position of C.

(c) On the same diagram, using ruler and compasses only, construct(i) the bisector of angle ABC , (ii) the perpendicular bisector of the line AB.

A

N

19

[3]

23 At School A, 160 pupils took an English Test.

The diagram below is the cumulative frequency curve for their results.

B

20

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

2 0 4 0 6 0 8 0 1 0 0M a r k s

0

Use the graph to find

(a) the interquartile range,

(b) the value of x , if 20% of the students scored x marks and above.

Answe

r

(a

)______________________ [1]

(b

)______________________ [1]

At another school, B, 120 pupils took the same English Test.

The diagram below is the box-and-whiskers plot for their results.

21

1 5 4 1 5 41 0 9 8 M a r k s

(c) Compare the test results for the two schools in two different ways.

Answer (c) _______________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________ [2]

~ END OF PAPER ~

Catholic High School2008 Mathematics Preliminary Examination 3

Paper 1 Answer Key1. 2000 m

2. (a) 70.2°C (b) 6.4− °C

3. 0, 1 and 2

22

4. 0,5.2,3

1,1232 −×

1232,0 ×

5. 20

6. 3

11or

3

15 −== xx

7. ( ) ( )bayx −− 54

8. 62

5

3 ba

9. (a) 5107.8 × (b) %49.7

10. (a) 113266 ××=

732168 3 ××=

(b) 6

(c) 28

11. (a) 12 −n (b) ( ) 212 −n (c) ( ) 232 +n

12. (a) 0 (b)12

7(c)

21

1

13. (a) 0.243 (b) 125%

14. (a)

AB

ε

36 92

5

7

4 8 1 01

0

(b) 1 (c) { }10,9,8,6,4,3,1,0

15. (a)

( ) ( )( )( )

8

62

362

100264

102124 22

==−

=−

=−+

=−+−

k

k

k

k

k

2

2

(b) ( 3 , 5) (c)4

3

(d)4

32

4

3 += xy or 1134 += xy

(e) 30 units2

16.

(a)

y

x

( - 2 , 1 )

- 3 - 1

- 3

(b) 2−=n (c) xy −= 2

17. (a) (i) 20 min (ii) 40min

(b)

0 2 0 4 0

D e p t h o f

w a t e r ( c m )

T i m e ( m i n u t e s ) 1 0 3 0

h2

h

18. (a) AAA Similarity

(b)(i) 9 : 16 (ii) 3 : 14

19. (a)(i) 5m/s2 (ii) 12.5 m/s (iii) 7.5m/s

23

(b) 0 2 4 6 8 1 0 2 01 2 1 4 1 6 1 8

2 0

4 0

6 0

8 0

1 0 0D i s t a n c e T r a v e l l e d

( m e t r e s )

T i m e ( t s e c o n d s )

1 2 0

1 4 0

1 6 0

1 8 0

20. (a) ( 5 , 0) (b) 4−=h or 10

21.

square. a is

., Since

.& rectangle, a is Since

.

　 triangletocongruent is triangleSince )c(

)Congruency (A.A.S. 　 triangletocongruent is triangle

line)straight aon angle(adjacent 90 　

rectangle) a is（90180

　　　　　　　)square a is(

) (a)part from ( 　　

, and esIn triangl (b)

＝　　　　

square) a is　　（90

rectangle) a is　　（90 (a)

DEFG

FEDGFGDEDGDE

FEDGFGDEDEFG

DGDE

CDGADE

CDGADE

DEFG 　DAE

ABCDCDAD

CDGADE

CDGADE

CDGADE　　　　　　　　　

CDGADGADGADE

CDGADGADCADGADEEDG

ADC DC　 　　EDG　　　　　

ABCDADC

DEFG EDG

∴===∴=

===

°=°−°=∠

=∠=∠

∠=∠∠+∠=∠+∠∴

∠+∠=∠∠+∠=∠∠∠∴

°=∠°=∠

23. (a) 16 (b) 66 or 67

(c) School B has a higher interquartile range at 39 as compared to that of School A at 16.

School A has a higher median at 55 as compared to that of School B at 41.