VS 2012 EM P1

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Class Register Number

Name

4016/01 12/4P2/EM/1

MATHEMATICS PAPER 1

Wednesday 15 August 2012 2 hours

VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL

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 pi , use either your calculator value or 3.142, unless the question requires the answer in terms of pi .

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.

2

VICTORIA SCHOOL 2012 12/4P2/EM/1

Mathematical Formulae

Compound interest

Total amount = 1100

nrP +

Mensuration

Curved surface area of a cone = rlpi

Surface area of a sphere = 24 rpi

Volume of a cone = 213

r hpi

Volume of a sphere = 343

rpi

Area of triangle ABC = 1 sin2

ab C

Arc length = r , where is in radians

Sector area = 212

r , where is in radians

Trigonometry

sin sin sina b c

A B C= =

2 2 2 2 cosa b c bc A= +

Statistics

Mean = fxf

Standard deviation = 22fx fx

f f

3


1 Particle N has a mass of 0.0345 micrograms and particle M has a mass of 6.73 nanograms.

(a) Express 0.0345 micrograms in grams. Give your answer in standard form.

Answer (a) grams [1]

(b) Find the difference in the masses of the two particles. Express your answer in nanograms.

Answer (b) nanograms [1]

2 In March 2012, the ratio of Jasons expenditure on food, transport and children is 3: 2: 5 respectively. In April 2012, his expenditure on children increases by 20% due to extra tuition fees. Jason would like to keep his total expenditure unchanged. Assuming that Jason is unable to reduce his transport cost, calculate the percentage reduction on his food expenditure required.

Answer % [2]

3 }{ : is an integer and 2 15 ,x x x = < }{ : is an even number ,P x x= }{ 2: 20 ,Q x x= }{ : is a factor of 24 .R x x=

(a) Find ( ).n Q

Answer (a) [1]

(b) List the element(s) in 'RP .

Answer (b) [1]

4


4 Mrs Lim took 18 minutes to drive to her office 15 km away from her house. She later travelled the same distance back but took only 13 minutes on her return trip.

(a) Express Mrs Lims driving speed to the office in kilometres per hour.

Answer (a) km/h [1]

(b) Calculate her average speed in kilometres per hour for her trip to the office and back.

Answer (b) km/h [1]

5 The first four terms of a sequence are 34, 26, 18, 10, .

(a) Find an expression, in terms of n, for the nth term of the sequence.

Answer (a) [1]

(b) If the pth term of the sequence is 254 , find the value of p.

Answer (b) p = [1]

6 y is inversely proportional to 3.x When x increases by 150 %, find the percentage decrease in y.

Answer % [2]

5


7 When written as the product of their prime factors

3

2 2 4

3 2

2 3 7,2 3 5 ,2 3 7 .

p

qr

=

=

=

Find the

(a) value of

the square root of q,

Answer (a) [1]

(b) greatest number that will divide p, q and r exactly,

Answer (b) [1]

(c) LCM of p, q and r, giving your answer as the product of its prime factors.

Answer (c) [1]

8 A map has a scale of 1 : n.

(a) A reservoir has an area of 28.1 km . It is represented by an area of 290 cm on the map. Find the value of n.

Answer (a) n = [2]

(b) The perimeter of the reservoir on the map is 10 cm. Find the actual perimeter of the reservoir, giving your answer in km.

Answer (b) km [1]

6


9 The temperature at 05 00 is 18 C. The temperature at 13 00 is 23 C.

(a) Find the difference between the two temperatures.

Answer (a) C [1]

(b) Assuming that the temperature rises at a steady rate, find the

(i) temperature at 11 00,

Answer (b)(i) C [1]

(ii) time when the temperature is 2.5 C.

Answer (b)(ii) [1]

10 (a) Given that 6 827

k

= , find the value of k.

Answer (a) k = [1]

(b) Simplify ( )2 6

2 41 2

4 5,

704a b b

aa b

expressing your answer in positive indices.

Answer (b) [2]

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11

The figure shows part of a n-sided polygon. The polygon has 4 of its interior angles of size .x

(a) It is given that the size of the interior angle, ,x is five times of its exterior angle. Find x.

Answer (a) x = [1]

(b) Each of the remaining interior angles of the polygon is 120 .

Find the value of n.

Answer (b) n = [2]

12 (a) Sketch the graph of ( )24 32.y x=

[2]

(b) Write down the coordinates of the minimum point of the curve.

Answer (b) ( , ) [1]

y

x O

x

x x

x

8


O

A (4, 3)B

C

y

x

13 The diagram shows a rectangle OABC. A and C lie on the y and x axis respectively. B is ( )4, 3 .

(a) Find the equation of the line AC.

Answer (a) [1]

(b) Show that 3 41 ,15 5

D

lies on the line AC.

Answer (b)

[1]

(c) Find the ratio : .AD DC

Answer (c) : [1]

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14 (a) Solve the inequality 12 139 6 95 3 65.x x x + < + Show your solution on the number line below.

Answer (a)

[3]

(b) Hence state the largest rational number which satisfies the inequality in (a).

Answer (b) [1]

15 Factorise fully

(a) 3 2 22

,

25 15 9a a b ab

+

Answer (a) [2]

(b) 2 2 10 25.y x x +

Answer (b) [2]

x

10


16 It is given that 5 3x

and 5.4 16.2y < where x and y are integers. Find the

(a) greatest value of ,x y

Answer (a) [1]

(b) greatest value of 2xy ,

Answer (b) [1] (c) least value of

2x

y,

Answer (c) [1] (d) least value of 2

1 2y x

+ .

Answer (d) [1]

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17

The diagram above is the speed-time graph for a cars journey. It travelled a total distance of 0.532 km in x seconds. Find the

(a) acceleration during the first 3 seconds,

Answer (a) 2m/s [1]

(b) speed when 1.35 s,t =

Answer (b) m/s [1] (c) value of x.

Answer (c) x = [2]

38

Speed (m/s)

x 3 10.5 Time (seconds)

20

15 0

12


18 A bag contains 10 balls of which 7 are blue and the rest are pink. A ball is drawn at random from the bag and its colour is noted. The ball is then removed and replaced with another ball of the other colour. A second ball is now drawn from the bag.

(a) Complete the tree diagram below. [2]

Answer (a)

(b) Find, in its simplest form, the probability that

(i) both balls drawn are blue,

Answer (b)(i) [1]

(ii) the second ball drawn is pink.

Answer (b)(ii) [2]

Blue

Pink

710

Blue

Pink

Blue

Pink

First ball Second ball

13


19 The dot diagram below shows the marks obtained by a group of 15 students in the English Preliminary Examination. The total marks for the examination was 80 marks.

(a) State the modal mark.

Answer (a) [1]

(b) State the median mark.

Answer (b) [1]

(c) A distinction is awarded to students who scored more than 80% in the examination. Find the percentage of students who scored a distinction.

Answer (c) % [1]

(d) The new mean mark will be 60 if the result of an additional student is considered. How many marks did the additional student obtain?

Answer (d) marks [2]

14


20 A company sells three types of hampers, A, B and C. The table below shows the number of items packed in each type of hamper.

Hamper Quantity

Number of cans of Abalone

Number of bottles of Chicken Essence

Number of boxes of Birds Nest

A 2 7 2 B 3 5 1 C 1 6 2

The selling prices of 1 can of Abalone, 1 bottle of Chicken Essence and 1 box of Birds Nest are $30, $3 and $15 respectively.

(a) Given that 2 7 23 5 11 6 2

=

P and

303 ,

15

=

Q

(i) evaluate ,=X PQ

Answer (a)(i) [1]

(ii) state what the elements of X represent.

Answer (a)(ii)

[1]

(b) During the Lunar New Year, the company receives 25 orders for Hamper A, 14 orders for Hamper B and 18 orders for Hamper C.

(i) Write down two matrices whose product shows the total sales of all the hampers ordered during the Lunar New Year.

Answer (b)(i) [2]

(ii) Evaluate this product.

Answer (b)(ii) [1]

15


P

Q

R

S

O

9 cm

6 cm

21 In the diagram, Q and S are points on both circles with centres P and O respectively. 9 cm and 6 cm.PQ OS= = QRS is a tangent to both circles. QS cuts PO at R.

(a) Given that 20 cm,QS = find SR.

Answer (a) cm [2] (b) Calculate

(i) QPR

in radians,

Answer (b)(i) QPR = radians [1]

(ii) the perimeter of the shaded region.

Answer (b)(ii) cm [1]

16


22

The diagram shows a circle PQRS with centre O. PQ and SR are produced to meet at U. QR and PS are produced to meet at T. It is given that 35 , 33 andQUR STR= =

.S RT x=

(a) Write down and PSR PQR in terms of x.

Answer (a) PSR = and PQR = [2]

(b) Form an equation in x and find .S RT

Answer (b) S RT = [2]

(c) Calculate .SOQ

Answer (c) SOQ = [2]

33

x

35

P

Q

R

S T

U

O

17


23

In the quadrilateral PQRS, the diagonals PR and QS intersect at T. It is given that PT ST= and .QT RT=

(a) Prove that triangles PQT and SRT are congruent.

Answer (a)

[3]

(b) Prove that triangles PTS and RTQ are similar.

Answer (b)

[2]

(c) Hence show that PS is parallel to QR.

Answer (c)

[1]

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.

Q

S P

R

T

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2012 Prelim 2 Mathematics Paper 1 Answer Key

1(a) 83.45 10 9(a) 41

1(b) 27.77 9(b)(i) 3124

2 1333

9(b)(ii) 09 00

3(a) 11 10(a) 3

3(b) { }10,14 10(b) 81172a

b

4(a) 50 11(a) 150

4(b) 25831

11(b) 8

5(a) ( )2 21 4n 12(b) ( )4, 32

5(b) 37 13(a) 3 34

y x= +

6 93.6 % 13(c) 2 : 3

7(a) 150 14(a) 1 13 533

x<

7(b) 6 14(b) 1533

7(c) 3 3 2 42 3 7 5 15(a) 2

5 3a b

a

8(a) 30 000 15(b) ( )( )5 5 y x y x+ +

8(b) 3

19


2012 Prelim 2 Mathematics Paper 1 Answer Key

16(a) 2 19(c) 2263

16(b) 0 19(d) 74

16(c) 25 20(a)(i) 11112078

16(d) 509768

20(a)(ii) The elements of X represent the total cost of each hamper A, B and C respectively.

17(a) 2123

20(b)(i) ( )25 14 18 and11112078

17(b) 17.1 20(b)(ii) ( )5859

17(c) 20.95 21(a) 8

18(a) 3 3 2 4 1, , , , 10 5 5 5 5

21(b)(i) 0.927

18(b)(i) 2150

21(b)(ii) 35.3

18(b)(ii) 1750

22(a) 33PSR x= + 35PQR x= +

19(a) 62 22(b) 56

19(b) 58 22(c) 112

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VS 2012 EM P1