Class Register Number

Name

4016/01 09/4P2/EM/1

MATHEMATICS PAPER 1

Friday 4 September 2009 2 hours

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VICTORIA SCHOOL

SECOND PRELIMINARY EXAMINATION

SECONDARY FOUR

Candidates answer on the Question Paper

READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid. Answer all the questions. 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 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 . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80.

Paper 1 consists of 12 printed pages, including the cover page. [Turn over

Secondary Four Second Preliminary Examination 2009 Victoria School

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Mathematical Formulae

Compound interest

Total amount = 1100

nrP +

Mensuration

Curved surface area of a cone = rl Surface area of a sphere = 24 r

Volume of a cone = 213

r h

Volume of a sphere = 343

r

Area of triangle = 1 sin2

ab C

Arc length = r , where is in radians Sector area = 21

2r , where is in radians

Trigonometry

sin sin sina b

A B= = c

C

A

2 2 2 2 cosa b c bc= +

Statistics

Mean = fxf

Standard deviation = 22fx fx

f f

Secondary Four Second Preliminary Examination 2009 Victoria School

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1 Evaluate 2466 991.58 using a calculator and round off your answer to (a) 2 significant figures, (b) the nearest million.

Answer (a) [1]

(b) [1]

2 When x is added to both the numerator and denominator of the fraction 23

, the result is

57

. Find x.

Answer x = [2]

3 Given that ( )32s r gh= , express h in terms of s, r and g.

Answer [2]

Secondary Four Second Preliminary Examination 2009 Victoria School

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4 Evaluate ( )2 10 6 3pp pqq .

Answer [2]

5 4 men take 3 hours to mow 1200 m2 of lawn. Given that all the men work at the same rate, calculate the time taken for 7 men to mow an area of 2900 m2.

Answer hours [2]

6 The first four terms of a sequence are given by 8, 64, 216, 512, . (a) Write down the next term of the sequence. (b) Find an expression, in terms of n, for the nth term of the sequence. (c) In another sequence, the first four terms are 10, 66, 218, 514, . By comparing this sequence with your answer in (b), write down an expression, in terms of n, for the nth term of the sequence.

Answer (a) [1]

(b) ..[1]

(c)..[1]

Secondary Four Second Preliminary Examination 2009 Victoria School

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7 E is the point ( )6 , 8 and F is the point ( )3, 4 . It is given that M is the mid-point of EF. Write down the position vector of M as a column vector.

Answer [2]

8 Four of the exterior angles of a 10sided polygon are xD each. The remaining exterior angles are ( each. )10x + D (a) Calculate x. (b) Hence write down the two possible values of an interior angle of the polygon.

Answer (a) x = [2]

(b) D and [2] D

9 Light travels at a speed of 300 000 000 m/s. (a) Express the speed of light in metres per nanosecond. (b) How many terametres does light travel in 1 year? Give your answer in standard form. (Take 1 year to be 365 days)

Answer (a) . m/ns [1]

(b) Tm [3]

Secondary Four Second Preliminary Examination 2009 Victoria School

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10 The following table gives the number of books borrowed by some students from the school library.

Number of books 1 2 3 4 5 6 x2+Number of students 12 9 11 5 3

(a) Write down the largest possible value of x given that the mode is 1. (b) Find the smallest possible value of x given that the median is 3. (c) Find the total number of students given that the mean is 3.2.

Answer (a) [1]

(b) [1]

(c) students [2]

11 (a) Express 2 8 3x x + in the form ( ) bax + 2 . (b) Sketch the graph of . 2 8y x x= +3

Answer (a) [1] (b)

[3]

y

x0

Secondary Four Second Preliminary Examination 2009 Victoria School

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12 (a) Solve 1 7 11 32 6

x x< + +2

.

(b) Write down all the factors of 30 which satisfy 1 7 11 32 6

x x< + +2

.

Answer (a) [3]

(b) [1]

13 At a hospital, the probability of a patient infected by the H1N1 virus is 0.2 . The probability that an infected person will be diagnosed correctly as carrying the virus is 0.9 . The probability that a non-infected person will be diagnosed wrongly is 0.01 .

(a) Complete the probability diagram shown in the answer space below.

[2] (b) A patient is chosen at random from the hospital. Find the probability that he will be diagnosed wrongly.

Infected with H1N1 virus

Not infected with H1N1 virus

Diagnosed correctly

Diagnosed wrongly

( ) 0.2

( ) ( ) ""

( )

( ) 0.9

( ) Diagnosed correctly

Diagnosed wrongly

Result of diagnosis Condition of patient

Answer (b) [2]

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14 The ratio of the areas of the bases of two geometrically similar cylinders, X and Y, is . The volume of X is 700 cm25: 49 3. (a) Calculate the volume of Y. (b) Another cylinder, Z, is half the height of cylinder X and has a radius which is 3 times that of cylinder X. Find the volume of cylinder Z.

Answer (a) cm3 [2]

(b) cm3 [2]

15 ACD is a right-angled triangle in which 7CD = cm. B is a point on AC such that cm. 15BC =

A

B

C D7 cm

15 cm

(a) Express, as a fraction, the value of tan ABD .

(b) It is given that 13

AB AC= . Calculate AD.

Answer (a) [1]

(b) cm [3]

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16 Solve 21 2 1 1

4 2 2x x x+ + = + .

Answer x = or [5]

17 The points A, B and C are ( ) , 2, 6 ( )2, 10 and ( )6 , 2 respectively. Find (a) the equation of the line BC, (b) the length of BC, (c) the coordinates of D if ABDC is an isosceles trapezium.

Answer (a) [2]

(b) units [2]

(c) [1]

( )2, 6 A

y

x

C ( )6, 2

O

B ( )2, 10

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18 A triangular army camp XYZ is such that 70XY = m, 95YZ = m and . Three watch towers are located at the corners X, Y and Z. It is completely fenced off along its perimeter.

65XYZ =

(a) Taking 1 cm to represent 10 m, construct a scale drawing of the army camp. (b) The headquarters must be situated such that it is equidistant from X, Y and Z. By suitable construction, find and label the position of the headquarters H. (c) The armoury is to be located such that it is 25m from the watchtower at Y and equidistant from the fences XY and YZ. By suitable construction, find and label the position of the armoury A.

Answer (a), (b) and (c)

[2][2][2]

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19 The diagram is the speed-time graph of a cars journey.

Speed(m/s)

Time (s)

0

v

10

16108 4

(a) Calculate the speed of the car at 3t = s. (b) If the total distance covered in the first 10 seconds is 92 m, find the value of v. (c) Find the retardation at t = 14 s. (d) On the axes below, draw the distance-time graph of the cars journey.

Answer (a) m/s [1]

(b) [2]

(c) m/s [1]2

(d)

Distance (m)

Time (s)

0 4 8 10 16 [4]

Secondary Four Second Preliminary Examination 2009 Victoria School

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20

A

B

P

C

O1.1 rad

In the diagram, A, B and C are points on a circle with centre O. The radius of the circle is 25 cm. BPC is an arc on another circle with centre A. It is given that rad. Find

1.1BAC = (a) the obtuse angle in radians, BOC (b) AB, (c) the area of the shaded portion, (d) the perimeter of the shaded portion.

Answer (a) BOC = radians [1]

(b) AB = cm [2]

(c) cm [4]2

(d) cm [2]

End of Paper

This document is intended for internal circulation in Victoria School only. No part of this document may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior permission of the Victoria School Internal Exams Committee.

Secondary Four Second Preliminary Examination 2009 Victoria School

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3

2009 Sec 4 E MATH Prelim 2 Paper 1 (Answers) 1 a) 2, 400, 000

b) 2, 000, 000 11 a) 2( 4) 1x

2 12

12

a) 3 157

x < b) 1, 2, 3, 5, 6, 10, 15

3 13 b) 0.028 38r shg=

4 73

1

p

14 a) 1920.8 cm 3

b) 3150 cm 3

5 147

hours 15

a) 715

b) 23.6 cm 6 a) 1000

b) 38nc) 38 2n +

16 -3.79 or 0.791

7 17 3 72

y x= 112

6

a)

b) 14.4 units c) ( 6 , -18 ) or ( 10, 2 )

8 a) 30 19 172

m/s a) b) 140 and 150 D D

b) 22 233

c) m/s 2

20 a) 2.2 radians 9 a) 0.3 m/ns b) 42.6 cm b) Tm 39.4608 10c) 245 cm 2d) 102 cm

10 a) 9 b) 1 c) 55

4 September 2009