27
[ Turn Over Class Index Number Name ANG MO KIO SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC MATHEMATICS Paper 1 4016/01 Wednesday 2 Sep 2009 2 hours Name of Setter: Mr Chio Kah Leong Candidates answer on the Question Paper. Additional Materials: Geometrical Instruments READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces on the top of this page. Write in dark blue or black pen in the spaces provided on the Question Paper. You may use a pencil for any diagrams or graphs. Do not use staple, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown in the space below that question. You are expected to use a scientific 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 the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . 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 This document consists of 20 printed pages 80

Amk Prelim 2009 Em1

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Page 1: Amk Prelim 2009 Em1

[ Turn Over

Class Index Number Name

ANG MO KIO SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC

MATHEMATICS Paper 1

4016/01 Wednesday 2 Sep 2009 2 hours Name of Setter: Mr Chio Kah Leong Candidates answer on the Question Paper. Additional Materials: Geometrical Instruments

READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces on the top of this page. Write in dark blue or black pen in the spaces provided on the Question Paper. You may use a pencil for any diagrams or graphs. Do not use staple, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown in the space below that question. You are expected to use a scientific 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 the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . 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

This document consists of 20 printed pages

80

Page 2: Amk Prelim 2009 Em1

[ Turn Over

Mathematical Formulae

Compound interest

Total amount = nrP

1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 4r2

Volume of a cone = hr 2

31

Volume of a sphere = 3

34 r

Area of triangle ABC = Cabsin21

Arc length = rθ, where θ is in radians

Sector area = 221 r , where θ is in radians

Trigonometry

Cc

Bb

Aa

sinsinsin

Abccba cos2222

Statistics

Mean = ffx

Standard deviation = 22

ffx

ffx

Page 3: Amk Prelim 2009 Em1

3

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

1 (a) The mass of a bacteria cell 111002.1 g can be written as k nanograms. Find k.

(b) Find the ratio of 2103.1 to 1105.6 . Give your answer in its simplest form.

Answer (a) k = ....………….………………… [1]

(b) …....………….. : .………….… [1]

2 The numbers 756 and 2352, written as a product of their prime factors, are

732756 32 and 24 7322352 .

Use these results to find

(a) the largest integer which is a factor of 756 and 2352,

(b) the smallest positive integer k for which 756k is a multiple of 2352,

(c) the smallest positive integer value of n for which 3 756n is a whole number.

Answer (a) ….....………….………………… [1]

(b) …....………….……………….… [1]

(c) ………………………………….. [1]

For Examiner's

Use

Page 4: Amk Prelim 2009 Em1

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

3 Solve the following equations

(a) x23

1

64

7xx + 3 = 0,

(b) yy

1255

2553

2

.

Answer (a) x = ……….……………………. [2]

(b) y = .……….……………………. [2]

Page 5: Amk Prelim 2009 Em1

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

4 Given that Y varies inversely as 42 x and that Y = 25 when x = 2, (a) express Y in terms of x,

(b) find the percentage increase in Y when x decreases by 25% from 2.

Answer (a) ….....………….………………… [2]

(b) …....………….………………. % [2]

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

5 Kelvin bought a crystal display for $180. He still made a percentage profit of 65%

despite offering a 40% discount to his customer. Calculate the selling price of the crystal

display before discount.

Answer $ ..………….…………………… [2] 6 The actual area of 2 km2 is represented on a map by an area of 50 cm2. Calculate

(a) the actual distance, in kilometres, represented by a length of 10.4 cm,

(b) the scale of the map, in the form n:1 .

Answer (a) …....………………………. km [2]

(b) 1 : …....……………………… [1]

For Examiner's

Use

Page 7: Amk Prelim 2009 Em1

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

7 In the diagram, a toy-car wheel of radius 6 cm is in contact with the horizontal ground at

P and touching the stair at R. Find in terms of ,

(a) the length of PQ,

(b) the height of the stair QR.

If = 3 , calculate

(c) the area of the sector OPR,

(d) the shaded area PQR.

Answer (a) ….....……………….……… cm [1]

(b) …....………………………. cm [1]

(c) …....……………………… cm2 [1]

(d) …....……………………… cm2 [2]

For Examiner's

Use

6 cm

P Q

R

O

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

8 In the diagram, the ratio of tan is 45 .

(a) Write down the coordinates of the point A.

(b) Find the equation of the line AB.

(c) Find the ratio of

(i) CABsin ,

(ii) CABcos .

Answer (a) (…...……..…. , …..……..….) [1]

(b) …....…………………………… [2]

(c)(i) …………………...…..……… [1]

(ii) …………………........……… [1]

A

B (2, 5)

y

x O

C

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

9 In the diagram (not drawn to scale), CEDCDB , CD = DE = 8 cm and

CE = 13 cm.

(a) Write down a second pair of equal lengths.

(b) Calculate the ratio of

(i) CD : CB ,

(ii) area of CBD : area of CDE .

Answer (a) …..…………...………..……….. [1]

(b)(i) …………………...…..……… [1]

(ii) …………………........……… [1]

B

C

E

13 cm

8 cm D

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10

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

10 In the diagram, EC is a tangent to the circle center O. Given that 63CED and ABD

is a straight line.

(a) Find, giving your reasons,

(i) CBA ,

(ii) reflex AOC ,

(iii) ACO ,

(iv) ECA .

(b) Is DE parallel to CA? State briefly a reason for your answer.

Answer (a)(i) …..…………...………..……..…° [1]

(ii) …..…………...………..……..…° [1]

(iii) …..…………...………..……..…° [1]

(iv) …..…………...………..……..…° [1]

Answer (b) ……………………………………………………………………………….

……………………………………………………………………………….. [1]

For Examiner's

Use

E

D

B A

O

C

63°

Page 11: Amk Prelim 2009 Em1

11

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

11 During a supermarket sale, a fruiterer sells two types of packages consisting of kiwi,

mango and strawberry.

Kiwi Mango Strawberry Package A 4 3 8 Package B 6 7 5

The cost of each kiwi, mango and strawberry is $0.90, $1.50 and $1.20 respectively.

A total of 50 Package A and 30 Package B were sold.

Given that

58

73

64

P ,

2.15.19.0

C and 3050T ,

(a) find PC.

(b) hence find TPC and explain what the answer represents.

Answer (a) PC = …….………..……………. [1]

(b) TPC = ………...…...…..……… [1]

Answer (b) ……………………………………………………………………………….

……………………………………………………………………………….. [1]

For Examiner's

Use

Page 12: Amk Prelim 2009 Em1

12

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

12 In the Venn diagrams below,

(a) write down the set notation that represents the shaded region,

Answer (a) …..…………...………..……….. [1]

(c) shade the region representing (B′ ∩ A)′.

Answer (b)

[1]

For Examiner's

Use

ε

A B

ε

A B

Page 13: Amk Prelim 2009 Em1

13

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

13 Two toy boats are geometrically similar and one is 3½ times longer than the other.

(a) Given that the height of the mast of the smaller boat is 12 cm, calculate the height

of the mast of the larger boat.

(b) The mass of the larger boat is 6.517 kg. What is the mass of the small boat in

grams?

Answer (a) ..…………...………..………...cm [2]

(b) .…………………...…..……......g [2]

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AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

14 Find the smallest integer value of x such that x + 3 15 4x 39.

Answer x = …..………...………..……….. [2]

15 The number of siblings that students in a class have is shown in the following table.

(a) If there are 40 students in the class, calculate the mean number of siblings.

(b) If the mode is 2, write down the range of values of x.

(c) Find the largest value of x if the median is 2.

Answer (a) …..…………...………..………… [2]

(b) ……………………...…..……….. [1]

(c) ……………………...…..……….. [1]

Number of siblings 0 1 2 3 4

Number of students 11 8 15 x 3

For Examiner's

Use

Page 15: Amk Prelim 2009 Em1

15

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

16 (a) Solve the equation 2231 tt .

(b) Factorise 4422 aba completely.

Answer (a) t = .…....…..………, .…..……..………… [2]

(b) …..……..….…….…...………………...... [2]

For Examiner's

Use

Page 16: Amk Prelim 2009 Em1

16

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

17 (a) Factorise 2542 2 xxy by using the completing the square method.

(b) Hence sketch the curve 2542 2 xxy in the axes below.

Label the y-intercept, x-intercepts and turning point clearly.

Answer (a) y = …………………...……… [2] Answer (b)

[2]

For Examiner's

Use

y

x 0

Page 17: Amk Prelim 2009 Em1

17

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

18 Solve the simultaneous equations

0242

32712

yx

yx

Answer x = ...…………...………..…………

y = ..…………………...…..………. [3]

19 The diagram shows two containers A and B of the same volume and height.

Water is poured into the containers at a constant rate until they are completely filled.

Given that the two containers are filled up completely at the same time, sketch on the

given axes, the height of water in the 2 containers against time as they are being filled.

Label your sketch clearly.

Answer

[2] time

height of water, h Maximum level of water

0

Container A Container B

h

Page 18: Amk Prelim 2009 Em1

18

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

20 (a) In the diagram, OPRQ is a parallelogram. OP p, OQ q and OT 3p.

(i) Express QT in terms of p and q.

(ii) Find the value of PTSofareaQRSofarea .

(iii) Hence, find the value of OPRQofarea

PSTofarea .

(b) Given a

013

, b

h9

, c

4h

and d

41

.

(i) Evaluate 3d a .

(ii) Given that b is parallel to c, calculate the values of h .

Answer (a)(i) …..…………...………..………… [1]

(ii) …..…………...………..………… [1]

(iii) .....…………...………..………… [1]

(b)(i) …..…………...………..………… [2]

(ii) h = …...……...………..………… [1]

For Examiner's

Use

T

S

Q

R

P

O p

q

Page 19: Amk Prelim 2009 Em1

19

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

21 A car travels at a constant speed of v m/s for 10 seconds and then accelerates uniformly

to a speed of 70 m/s before coming to a stop at 35 seconds.

(a) Calculate the value of v if the distance travelled in the first 25 seconds is 1470 m.

(b) Calculate the speed of the car, in m/s, when time is 20seconds.

(c) Calculate the acceleration of the car in the last 10 seconds.

(d) On the axes in the answer space below, sketch the acceleration-time graph of the

car for the 35 seconds. Label the vertical axis clearly.

Answer (a) …..…………...………..……...m/s [2]

(b) ……………………...…..…….m/s [2]

(c) ……………………...…..……m/s2 [1] (d)

[1]

For Examiner's

Use

Speed in m/s

Time (t) in seconds 10 25 35

v

70

0

Acceleration in m/s2

Time in seconds 10 25 35 0

Page 20: Amk Prelim 2009 Em1

20

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

22 The diagram shows a radio station A in a Town. Using a scale of 1 cm to 10 km, draw and label

(a) housing estate B located 90 km apart, at a bearing of 320 from radio station A,

(b) industrial estate C located 120 km west of radio station A.

Construct

(c) the angle bisector of BAC,

(d) the perpendicular bisector of AC. [4]

For Examiner's

Use END OF PAPER

N

A

Page 21: Amk Prelim 2009 Em1

2

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

Sec 4E / 5N Preliminary Exam 2009 E-Maths Paper 1

Answer Marking Scheme

1(a)

1(b)

21002.10102.0 or 1 : 50

B1

B1

2(a) 2(b) 2(c)

847322 HCF

2872732732

7327567322352756

2

23432

234

234

kk

kandofLCM

9872756

,756

2

3

nnumbercubeabemustn

numberwholeaisnIf

B1

B1

B1

3(a) 3(b)

4520

01218720232372

03232

723

1

0364

723

1

xx

xxxx

xx

x

xx

x

321

33122

55

55 3

31

22

y

yy

yy

M1

A1

M1

A1

4(a)

4200200825

,25,2

.tan,4

2

2

xY

kYxWhen

tconsaiskwherex

kY

M1

A1

Page 22: Amk Prelim 2009 Em1

3

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

4(b)

%28

%10025

2532%

3245.1

2005.1275.0

%,25

2

Yinincrease

Ynew

xnewbydecreasesxWhen

M1

A1

5

Selling price after discount 297$65.1180$

Selling price before discount

495$

%10060297$

M1

A1

6(a) 6(b)

kmcm

kmcmkmcmkmcm

08.25

4.104.10

1:51:252:50

22

22

00020:100020:1

51:1

cmcm

kmcm

M1

A1

B1

7(a) 7(b) 7(c) 7(d)

sin6 cos66

2

2

2

2

8.186

36

2121

sec

cmcm

r

OPRtorofArea

B1 B1

B1

Page 23: Amk Prelim 2009 Em1

4

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

2

2

53.4

62

327

63

sin63

cos66621

621

cm

cm

PQRQOP

PQRareaShaded

M1

A1

8(a) 8(b) 8(c)(i) 8(c)(ii)

0,22

45

205

Ax

x

105425

45

2455

xyor

xy

xy

415sinsin CAB

414coscos CAB

B1

M1

A1

B1

B1

9(a) 9(b)(i) 9(b)(ii)

DB = BC

8:13:8

13.8

.

CBCDCDCE

CBCD

cmDECDCEDtosimilarisCDB

169:64..169642

eiCDCB

CDEofAreaCBDofArea

B1

B1

B1

10(a)(i) 10(a)(ii)

117.63180

63linestraonsadjCBA

segmentsameinsDECDBC

.22342117

circumatcentreatAOCreflex

B1

B1

Page 24: Amk Prelim 2009 Em1

5

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

10(a)(iii) 10(a)(iv) 10(b)

isosofsbase

AOC

272

234360180

63tan2790 radiusECA

Yes. Since ECADEC , they form alternating angles i.e. DE // AC.

B1

B1

B1

11(a) 11(b)

90.2170.17

PC

1542TPC

It represents the total amt. of money collected from the sale of 50 package A and 30 package B.

B1

B1

B1

12(a) 12(b)

''''''

,

BAorABBAorAB

followingtheAccept

B1

B1

13(a) 13(b)

cmh

h

4215.3

12

gkgmass

mass

smaller

smaller

152152.0

3438

72

5.31

517.6

33

M1

A1

M1

A1

ε

A B

Page 25: Amk Prelim 2009 Em1

6

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

14

.5int4.26

64.2244125

394154153

isxegerSmallestx

xandxxandx

xandxx

M1

M1

A1

15(a) 15(b) 15(c)

If total = 40, x = 3.

.48.1475.148.1

475.140

343315281110

tooffroundingbeforeshowstudentsifacceptOnly

Mean

150140 xorx

30

M1

A1

B1

B1

16(a) 16(b)

134

0143043

22312

tort

tttt

tt

baba

ba

aba

222

4422

22

M1

A1

M1

A1

17(a)

2712

22712

225

22

2222

22522

2

2

222

2

x

x

xx

xxy

M1

A1

Page 26: Amk Prelim 2009 Em1

7

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

17(b)

B1 – Correct shape of curve (minimum) B1 – Label the y-intercept, x-intercepts and turning point.

18 15,6 yx M2 + A1

19

B2

20(a)(i) 20(a)(ii) 20(a)(iii)

qp 3

41

32

B1

B1

B1

-2.67

-25

4.67

(1, -27)

time

height of water, h Maximum level of water

0

A

B

Page 27: Amk Prelim 2009 Em1

8

AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over

20(b)(i) 20(b)(ii)

2012163

1216

3

22

ad

ad

64

9

h

hh

M1

A1

B1

21(a) 21(b) 21(c) 21(d)

smv

vv

/54

147015702110

sms

s

/3264

155470

1054

- 7 m/s2

M1

A1

M1

A1

B1

B1

Acceleration in m/s2

Time in seconds 10 25 35 0 1511

-7