Amk Prelim 2009 Em1

Class

Index Number

Name

ANG MO KIO SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC MATHEMATICS Paper 1 Wednesday Name of Setter:

4016/01 2 Sep 2009 Mr Chio Kah Leong

2 hours

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

80 This document consists of 20 printed pages [ Turn Over

Mathematical Formulae

Compound interest r   Total amount = P1    100 

n

Mensuration Curved surface area of a cone = rl Surface area of a sphere = 4 r2 Volume of a cone =

1 2 r h 3

Volume of a sphere = Area of triangle ABC =

4 3 r 3

1 ab sin C 2

Arc length = rθ, where θ is in radians Sector area =

1 2 r  , where θ is in radians 2

Trigonometry a b c   sin A sin B sin C

a 2  b 2  c 2  2bc cos A Statistics Mean =

Standard deviation =

 fx f  fx 2   fx     f f 

2

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11

(a)

The mass of a bacteria cell 1.02  10

g can be written as k nanograms. Find k.

(b)

Find the ratio of 1.3  10 2 to 6.5  10 1 . Give your answer in its simplest form.

For Examiner's Use

Answer (a) k = ....………….………………… [1] (b) …....………….. : .………….… [1] 2

The numbers 756 and 2352, written as a product of their prime factors, are 756  2 2  3 3  7 and 2352  2 4  3  7 2 .

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] AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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3

Solve the following equations (a)

(b)

1 7x  + 3 = 0, 3  2x 4x  6

5 2 y  25 3

5

 125 y .

Answer (a) x = ……….……………………. [2] (b) y = .……….……………………. [2] AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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

2



Given that Y varies inversely as x  4 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]

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

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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

For Examiner's Use

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 1 : n .

Answer (a) …....………………………. km [2] (b) 1 : …....……………………… [1]

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

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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  = (c) (d)

 , calculate 3

For Examiner's Use

O

the area of the sector OPR,

6 cm



the shaded area PQR.

P

R

Q

Answer (a) ….....……………….……… cm [1] (b) …....………………………. cm [1] (c) …....……………………… cm2 [1] (d) …....……………………… cm2 [2]

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

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8

5 In the diagram, the ratio of tan  is . 4 y

B (2, 5)

 C

A

x

O

(a)

Write down the coordinates of the point A.

(b)

Find the equation of the line AB.

(c)

Find the ratio of (i)

sin CAB ,

(ii)

cos CAB .

Answer (a) (…...……..…. , …..……..….) [1] (b) …....…………………………… [2] (c)(i) …………………...…..……… [1] (ii) …………………........……… [1] AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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9

In the diagram (not drawn to scale), CDB  CED , CD = DE = 8 cm and CE = 13 cm.

(a)

Write down a second pair of equal lengths.

(b)

Calculate the ratio of

D

(i)

CD : CB ,

(ii)

area of CBD : area of CDE .

8 cm E

B

13 cm

C

Answer (a) …..…………...………..……….. [1] (b)(i) …………………...…..……… [1] (ii) …………………........……… [1]

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

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10

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

For Examiner's Use

is a straight line. E (a)

Find, giving your reasons, (i)

CBA ,

(ii)

reflex AOC ,

(iii) ACO , (iv)

(b)

 ECA .

63°

A B

O

D C

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]

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

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11

During a supermarket sale, a fruiterer sells two types of packages consisting of kiwi, mango and strawberry. Package A Package B

Kiwi 4 6

Mango 3 7

For Examiner's Use

Strawberry 8 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.  0.9     4 3 8 Given that P    , C   1.5  and T  50 30 ,  6 7 5  1 .2   

(a)

find PC.

(b)

hence find TPC and explain what the answer represents.

Answer (a) PC = …….………..……………. [1]

(b) TPC = ………...…...…..……… [1] Answer (b) ………………………………………………………………………………. ……………………………………………………………………………….. [1]

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

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12

For Examiner's Use

In the Venn diagrams below,

(a)

write down the set notation that represents the shaded region,

ε A

B

Answer (a) …..…………...………..……….. [1]

(c)

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

Answer (b)

ε A

B

[1]

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

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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] AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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14

For Examiner's Use

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. Number of siblings

0

1

2

3

4

Number of students

11

8

15

x

3

(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] AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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16

(a)

Solve the equation t  1 3t  2  2 .

For Examiner's Use

(b) Factorise a 2  b 2  4 a  4 completely.

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

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

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17

(a)

Factorise y  2 x 2  4 x  25 by using the completing the square method.

(b)

Hence sketch the curve y  2 x 2  4 x  25 in the axes below.

For Examiner's Use

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

Answer (a) y = …………………...……… [2]

Answer (b) y

0

x

[2]

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

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18

Solve the simultaneous equations

12 x  27  3 y x  2 y  24  0

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.

h

Container A

Container B

Answer height of water, h

Maximum level of water

[2] 0 AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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20

(a)

For Examiner's Use

In the diagram, OPRQ is a parallelogram. OP  p, OQ  q and OT  3p. Q q O

R S p P T

(i)

Express QT in terms of p and q.

(ii)

Find the value of

area of QRS . area of PTS

(iii) Hence, find the value of

(b)

area of PST . area of OPRQ

13  9  h   1 Given a    , b    , c    and d    . 0 h  4  4 (i)

Evaluate 3 d  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]

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

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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.

For Examiner's Use

Speed in m/s 70 v

0

10

25 35 Time (t) in 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)

Acceleration in m/s2

0

10

25

35

Time in seconds

[1] AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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22

For Examiner's Use

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]

N

A

END OF PAPER AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01

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

Marking Scheme

1(a) 1(b)

0.0102 or 1.02  10 2 1 : 50

B1 B1

2(a)

HCF  2 2  3  7  84

B1

2(b)

LCM of 756 and 2352  2 4  3 3  7 2  2 4  33  7 2

756k

2 2  33  7  k  2 4  3 3  7 2  k  2 2  7  28

2(c)

If 3 756n is a whole number,  756n must be a cube number  n  2  7 2  98

3(a)

B1

1 7x  30 3  2x 4x  6 1 7x  30 3  2 x 23  2 x   2  7 x  32 3  2 x   0

B1

M1

 2  7 x  18  12 x  0 20  5 x x4 3(b)

52 y  52 5

A1

 53 y

1 3

 2y  2  y 1 4(a)

Y 

1  3y 3

2 3

M1 A1

k

, where k is a cons tan t. x 4 When x  2, Y  25, k  258  200 2

 Y 

200 x2  4

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

M1 A1

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4(b)

When x decreases by 25%, new x  0.752   1.5 200  new Y   32 1.5 2  4 32  25 % increase in Y   100% 25  28%

5

6(a)

A1

Selling price after discount  $180  1.65  $297

M1

Selling price before discount $297   100% 60  $495

A1

50 cm 2

: 2 km 2

25 cm 2 : 1 km 2  5 cm : 1 km  10.4 cm  6(b)

M1

10.4  2.08 km 5

M1 A1

1 km 5 1 cm : 20 000 cm  1 : 20 000

B1

7(a) 7(b)

6 sin  6  6 cos 

B1 B1

7(c)

Area of sec tor OPR

1 cm :

 

1 2 1 2

r 2

  3

6 2 

 6 cm 2

B1

 18.8 cm 2

7(d)

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

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Shaded area PQR 1 OP  RQ PQ   6 2 1      6  6  6 cos  6 sin   6 2 3  3  27 3     6  cm 2  2  2  4.53 cm

M1



8(a)

50 5  2x 4 x  2 A   2, 0 

B1

5 x  2 4 5 5 y  x 4 2 or 4 y  5 x  10

8(b)

y5

8(c)(i)

sin CAB  sin  

M1

A1

5

B1

41 4

8(c)(ii)

cos CAB   cos   

9(a)

DB = BC

9(b)(i)

CDB is similar to CED. CD  DE  8 cm.

10(a)(i)

B1

B1 2

Area of CBD  CB  64    Area of CDE  CD  169 i.e. 64 : 169

B1

DBC  DEC  63 s in same segment  CBA  180  63 adj s on a str . line   117

10(a)(ii)

B1

41

CD CE 13   CB CD 8  CD : CB  13 : 8 9(b)(ii)

A1

B1

reflex AOC  117  2  234 at centre  2 at circum.

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

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10(a)(iii)

10(a)(iv)

10(b)

180  360  234 2  27 base s of isos  

AOC 

B1

tan  radius 

ECA  90  27  63

B1

Yes. Since DEC  ECA , they form alternating angles i.e. DE // AC.

B1

11(a)

 17.70   PC    21.90 

B1

11(b)

TPC  1542

B1

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

B1

12(a)

Accept the following , B  A' or A' B B' A ' or  A  B' '

12(b)

B1

ε A

B B1

13(a)

13(b)

h 3 .5  12 1  h  42 cm

mass smaller 6.517 mass smaller

M1 A1 3

3

8  1  2      343  3.5  7  0.152 kg  152 g

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

M1

A1

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14

x  3  15  4 x 5 x  12 x  2.4

15(a)

and and

and

15  4 x  39

 4 x  24 x  6

  6  x  2.4 Smallest int eger x is  5.

M1 M1 A1

If total = 40, x = 3. 011  18  215  33  43 Mean  40  1.475

M1 A1

 Only accept 1.48 if students show 1.475 before     rounding off to 1.48.  15(b)

0  x  14 or 0  x  15

B1

15(c)

30

B1

16(a)

t  13t  2  2 3t 2  t  4  0

3t  4t  1  0 t

16(b)

4 3

or t  1

A1

a 2  b 2  4a  4  a  2   b 2  a  2  b a  2  b  2

17(a)

M1

25   y  2 x 2  2 x   2  2 2    2    2  25   2 x 2  2 x        2   2   2   27   2  2  x  1   2   2 x  1  27 2

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

M1 A1

M1 A1

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17(b)

4.67

-2.67 -25 (1, -27)

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

18

M2 + A1

x  6, y  15

19 height of water, h

Maximum level of water

B

B2 A

0

time

20(a)(i)

3p  q

B1

20(a)(ii)

1 4

B1

2 3

B1

20(a)(iii)

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

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20(b)(i)

20(b)(ii)

21(a)

21(b)

21(c)

21(d)

  16   3d  a    12 

M1

3d  a  16 2  12 2  20

A1

9 h  h 4 h  6

B1

1 v  7015  1470 2 v  54 m / s s  54 70  54  10 15 2 s  64 m / s 3

M1

10v 

A1 M1 A1

- 7 m/s2

B1

Acceleration in m/s2

B1

1 1 15 0

10

25

35

Time in seconds

-7

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

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