Class Register Number

Name

4016/02 09/4P2/EM/2

MATHEMATICS PAPER 2

Tuesday 08 September 2009 2 hours 30 minutes

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

Additional Materials: Answer Paper Graph Paper (1 sheet)

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 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 100.

Paper 2 consists of 9 printed pages, including the cover page. [Turn over

Secondary Four Second Preliminary Examination 2009 Victoria School

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 = 34

3r

Area of triangle ABC = 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

CA

2 2 2 2 cosa b c bc= +

Statistics

Mean = fxf

Standard deviation = 22fx fx

f f

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Secondary Four Second Preliminary Examination 2009 Victoria School

1 A universal set and its subsets A and B are given by { }: is an integer such that 1 15x x x = , { }: is a prime numberA x x= , { }: is a multiple of 3B x x= .

(a) Draw a Venn Diagram showing , A and B and place each of the members 1 to 15 in the appropriate part of the diagram. [3] (b) State the value of n ( )A B . [1] (c) List the elements contained in the set ( ) 'A B . [1] (d) Describe in words, the elements contained in the set 'A B . [1] 2 The price and the number of sets of LCD television sold during a recent four-day electronic fair is given in the following table.

Size of TV Day 32 inches 36 inches 42 inches 50 inches

Day 1 20 15 10 5 Day 2 10 18 15 2 Day 3 12 20 5 8 Day 4 8 14 12 9

Selling Price $700 $1000 $1200 $2500 (a) Write down two matrices X and Y such that matrix multiplication will give the total sales for each day of the fair. [2] (b) Hence evaluate this matrix multiplication. [2] (c) Write down a matrix Z such that matrix multiplication of this matrix with your answer in (b) will give the total sales for the entire duration of the fair. Evaluate the total sales at the fair. [2] 3 The table shows the number of goals scored in each soccer match played by Victoria Schools soccer team A in a recent tournament.

Number of goals scored in each match 0 1 2 3 4 5 Number of matches 5 8 2

(a) Calculate the mean number of goals scored per game. [2] (b) Calculate the standard deviation. [2] (c) Victoria Schools soccer team B has a mean of 2.11 goals scored per game and a standard deviation of 2. Which team should be selected to represent the school in an important game if the coach will like to select a team which is competitive as well as consistent in scoring goals. Justify your answer. [2]

3

Secondary Four Second Preliminary Examination 2009 Victoria School

4 (a) The heights of 80 pupils in Littlepool Primary was measured. The cumulative frequency curve below shows the distribution of their heights. Use the curve to estimate

0

10

20

30

40

50

60

70

80

90

90 100 110 120 130 140 150Height (cm)

Cum

ulat

ive

Freq

uenc

y

(i) the range, [1] (ii) the median height, [1] (iii) the interquartile range. [2] Pupils who are above a certain height will be selected to perform at the National Day Parade (NDP). The selection results revealed that 20% of the pupils were tall enough. (iv) Find the least height for a pupil to be selected. [2] (b) The box and whisker diagram below illustrates the heights in cm obtained from 80 pupils in Kiddenham Primary.

130 150140 100 110 12090

Height (cm)

Students from Kiddenham Primary were also selected to perform at the NDP based on the same height criteria found in (a)(iv). The newspapers reported that Kiddenham Primary had more students performing at the NDP than Littlepool Primary. Is the newspapers report accurate? Give a reason for your answer. [2]

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Secondary Four Second Preliminary Examination 2009 Victoria School

5 (a) In a particular year, Samuel received a salary of $7 250 per month. He donated 1% of his income to charitable organisations. He was entitled for an annual relief amount of $7 000 for looking after his parent and a handicapped child. Taxable income refers to annual gross income less all donations and relief for the same year. (i) Calculate Samuels taxable income for the year. [2] (ii) Taxable income is taxed at the following rate:

a flat rate of $900 on the first $40 000 earned and

a rate of 8.50% on the amount above $40 000. Calculate the amount of income tax that Samuel had to pay for the year. [2] (b) Kara invested a sum of money with a finance company which earned a compound interest of 4.5% per annum. If Kara had a total of $23 653.28 with the finance company at the end of four years, calculate her initial investment amount, correct to the nearest cents. [2] (c) (i) Brian went to Brisbane for a holiday. The rate of exchange between Australian dollars (A$) and Singapore dollars (S$) was A$1 = S$1.15. The money changer charged a 1.5% commission for the transaction. How much Singapore dollars did Brian have to pay to the money changer if he bought A$3 000? [2] (ii) In Brisbane, Brian bought the latest iPhone at A$879.75. The shop where he bought the phone from had sold the phone to him at a profit of 15%. Calculate the amount in Australian dollars that the shop had paid for the phone. [2] (iii) When Brian returned from his holiday, he discovered that the retail price of the iPhone in Singapore was S$1 199. Calculate his percentage savings. [2]

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Secondary Four Second Preliminary Examination 2009 Victoria School

6 (a) In the diagram, and aOA =JJJG 3bOB =JJJG .

A

O

B

a

3b

(i) Find, in terms of a and b, the vector AB

JJJG. [1]

(ii) C is a point such that A, B and C are collinear. It is given that : 3AB BC =JJJG JJJG :1 . Find, in its simplest form, in terms of a and b, the two possible vectors AC

JJJG.

[2] (b) The figure shows a parallelogram PQRS. T is a point on PQ such that ST = PS. The lines SQ and RT intersect at U.

S

P

U

T Q

R

(i) Show that is congruent to PSQ STR . [2] (ii) Show that is similar to STU RQU . [2] It is given that : 4PT TQ :1= . Showing your working clearly, find the ratio (iii) area of triangle TQU : area of triangle RSU, [1] (iv) area of triangle PTS : area of triangle STR, [1] (v) area of triangle PTS : area of parallelogram PQRS. [2]

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Secondary Four Second Preliminary Examination 2009 Victoria School

7 The diagram shows a circle, ABCD, with centre O. PAX and PBY are tangents to the circle. It is given that 108XPY = , 36DAO = and 59ADC = . (a) Find . [2] BAO

O

A

B

C

D

108 36

P

X

59

Y

(b) Find . [3] BOC (c) Find . [2] OCD (d) Show that BOD is a diameter. [2]

W X

Y

Z North

135120

34

18

10

280

8

The diagram shows the locations of four towns W, X, Y and Z. The bearing of X from W is . It is given that , 280 120YXZ = 135WXY = , 34WY = km, km and km.

18WX =10XZ =

(a) Calculate the bearing of Y from W. [2] (b) Calculate the distance between Town W and Town Z. [3] (c) There is a hill at Town Z. The angle of elevation of the top of the hill from Town X is 8 . Calculate the height of the hill in metres. [2] (d) A lorry is moving along WY. Find the nearest distance of the lorry from Town X. [2]

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Secondary Four Second Preliminary Examination 2009 Victoria School

9 (a) (i) Factorise completely 24 4b bc ac ab + . [2]

(ii) It is given that 2 24 4x xy y 0+ + = . Find the value of xy

. [2]

B

C

A

D

( )3x +( )3 7x

( )1x

(b) In the diagram, ABCD is a trapezium in which AD is parallel to BC and . It is given that cm,

90ABC = ( )3 7AD x= ( )1AB x= cm and ( 3BC x )= + cm.

(i) Write down, in terms of x, an expression for the area of the trapezium. [2] (ii) It is given that the area of the trapezium is 20 cm . Form an equation in x 2 and show that it reduces to 2 2 9x x 0 = . [1] (iii) Solve the equation 2 2 9x x 0 = , giving the solutions correct to 2 decimal places. Hence, calculate the perimeter of the trapezium. [4] 10 Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation 24 2

2xy

x= + + . Some corresponding

values are given in the following table.

x 0.5 1 2 3 4 5 6 y 18.3 6.5 4 p 4.3 4.7 5.1

(a) Find the value of p. [1] (b) Using a scale of 2 cm to represent 1 unit, draw a horizontal xaxis for . Using a scale of 2 cm to represent 2 units, draw a vertical yaxis for .

0 6x 0 2y 0

On your axes, plot the points given in the table and join them with a smooth curve. [3] (c) Use your graph to find the range of values of for 0.75y 4.5x . [2] (d) By drawing a suitable straight line on your graph, find the solution of 2

4 2 0x

= . [3]

(e) By drawing a tangent, find the gradient of the curve 24 2

2xy

x= + + at the point

where 1x = . [2] 8

Secondary Four Second Preliminary Examination 2009 Victoria School

11 Diagram I

5

O

A B

C

D30

Diagram II

x

5 Diagram I shows a solid consisting of two hemispheres of radius 5 cm and a cylinder of radius 5 cm and height x cm joined together. (a) Find the volume of the solid, leaving your answer in terms of x and . [2]

(b) It is given that the volume of the solid is 250 . Show that 133

x = . [1] (c) Find the height of the solid. [1] The solid is melted completely and recast to form a prism as shown in Diagram II. The cross-section of the prism is a sector of a circle with centre O and radius 5 cm. It is given that . 30AOB = Taking to be 3.142, find (d) the area of the sector OACB, [2] (e) BD, [2] (f) the total surface area of the prism. [3]

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.

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Secondary Four Second Preliminary Examination 2009 Victoria School

Secondary Four Second Preliminary Examination 2009 Answer Key

Q1(a)

(b)

(c)

(d)

( ) 1n A B =

( ) ' {1, 4,8,10,14}A B = Elements that are prime numbers and indivisible by 3

Q6(a)(i)

(ii)

(b)(iii)

(iv)

(v)

AB =JJJK 3b a ACJJJK

= 23

(3b a) or 43

(3b a)

1 : 25

4 : 5

2 : 5

A B 2 5 7 11 13

3 6 9 12 15

1 4 8 10 14

54BAO = Q(

(

7(a)

b)

c)

46BOC = 23OCD =

Q2(a) 20 15 10 510 18 15 212 20 5 88 14 12 9

X

=

700100012002500

Y

=

53500

480005440056500

(b) 257.0Q8(a) (b) 22.7 km (c) 1410 m ( )1 1 1 1Z = , ( )212400 (d) 7.04 km (c) Q3(a)

(b)

(c)

Mean = 2.1

S.D. = 1.31

Team A. Its standard deviation is smaller thus its more consistent in its scoring.

( )( )4b a b c+ Q9(a)(i)

12

xy= (ii)

22 4x x 2 + (b)(i)

(ii)

4.16 or 2.16( )x x rej= = Perimeter = 19.4 cm

Q4(a)(i) Range = 46 cm

(ii) Median = 118 cm

(iii) Interquartile range = 11 cm Q10(a) p = 3.9 (iv) Height criteria = 125.5 cm 3.85 9.5y (c) (b) Newspaper is accurate. There are

more than 25% of the students in Kiddenham Pri above 125.5 cm

(d) 1.45x = (e) Gradient = 7.5

Q5(a)(i) Taxable income = $79130 Q11(a) 225 1663

x + (ii) Tax = $4226.05

1(b) P = $19834.73 (c) Height = 13 cm 3 (c)(i) S$3501.75 (d) 6.55 cm2

(ii) A$765 (e) 120 cm (iii) 15.6% 1530 cm2 (f)

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Secondary Four Second Preliminary Examination 2009 Victoria School

42xy = +

2

4 22xy

x= + +

11

08 September 2009