Anglo-Chinese School (Independent)

PRELIMINARY EXAMINATION 2012 YEAR FOUR EXPRESS

MATHEMATICS PAPER 2

4016/02

Wednesday 1st August 2012 2 hours 30 minutes

Additional Materials: Answer Paper (8 sheets) Graph Paper (1 sheet) READ THESE INSTRUCTIONS FIRST Write your index 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 staples, 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. Calculators should be used where appropriate. 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. This question paper consists of 10 printed pages. [Turn over

Mathematical Formulae

Compound Interest

Total amount = 100

r 1

+n

P

Mensuration Curved surface area of a cone = rl

Surface area of a sphere = 4 2r

Volume of a cone = hr 231

Volume of a sphere = 334 r

Area of a triangle = 21 absin C

Arc length = r , where is in radians

Sector area = 221 r , where is in radians

Trigonometry

Cc

Bb

Aa

sin

sin

sin==

a 2 = b 2 + c 2 2bc cos A

Statistics

Mean =

ffx

Standard deviation = 22

ffx

ffx

3

Answer all the questions. 1 Mr Wong is paid $5.00 per hour for normal working hours. He is considered to have worked overtime

if he works more than 217

hours on weekdays and more than 214 hours on Saturdays. The overtime

rate is 211 times the normal rate. On Sundays, Mr Wong is paid $5x per hour. During the first week of

August, Mr Wong is paid $165 for overtime after he has worked the following number of hours:

Mon Tue Wed Thur Fri Sat Sun

10 12 9 10 7 12 712 4

(a) Find x. [2] (b) If his deduction for CPF is 20% of his earnings, calculate his take home salary for that week. [2] (c) The amount that Mr Wong earns for overtime in the first week of August is 25% more than the

second week, calculate the amount he is paid for overtime in the second week. [2] (d) Mr Wong decides to invest $10 000 in a bank at an interest rate of R % compounded quarterly

for a period of 2 years. Calculate the value of R if his investment gives him a 10% return? [2]

2 Mr Chan decided to spend $54 on buying cutlery. A spoon and a fork cost $7.50 in total. If he spent all

his money on spoons alone, he could buy 6 more spoons than he could if he spent all the money on

forks.

Taking the cost of a fork as $x, write down expression in terms of x for (a) the cost of a spoon, [1] (b) the number of spoons which could be bought for $54. [1] (c) Hence, write down an equation which x must satisfy, and show that it reduces to

0135212 2 =+ xx . [3]

(d) Solve this equation by completing the squares. Hence state the cost of a spoon. [3]

4

3 A bakery sells two varieties of cupcakes, Dark Chocolate and Red Velvet. Each variety is sold in packets of three different sizes, small, medium and large of different prices. The sales in the first two weeks of July are given in the table below.

First Week Second Week

Size Small Medium Large Small Medium Large

Cost per cupcake $1.00 $1.50 $2.20 $1.00 $1.50 $2.20

Number of

Dark Chocolate sold 25 34 20 20 46 15

Number of

Red Velvet sold 18 26 30 20 30 40

The information for the first week of Julys sales can be represented by the matrix,

=

302618203425

P

and the cost of each cupcake for each size can be represented by the matrix

=

2.25.1

1A . The

information for the second week sales can be represented by the matrix Q.

(a) Write down the matrix Q. [1]

(b) Calculate S = P + Q. [1]

(c) Describe what is represented by the elements in S. [1]

(d) Calculate R = 21 SA. [2]

(e) Describe what is represented by the elements of R. [1]

(f) Calculate and describe what is represented by T = ( )11 SA. [2]

5

4 The diagram shows two circles intersecting at D and B. Points A, B, C, E and D form a triangle as shown. Point X is the mid-point of the minor arc AB.

(a) Prove that ACE is similar to ADB . [2]

Given that angle BXD = 38 and angle BDE = 92, calculate (b) BCE , [1] (c) CBD , [2] (d) XAD . [3]

5 Answer the whole of this question on a sheet of graph paper.

During a journey, the petrol consumption, C in litres of petrol of a car for every 100 kilometres is

related to the speed of the car, S (km/h) and is given by the equation 5.010001.0 2 += SSC for 8010 S . The table below shows some corresponding values of S and C.

S 10 20 30 40 50 60 70 80 C 10.5 4.0 3.6 4.1 4.9 a 6.7 7.7

(a) Calculate a. [1] (b) Taking 2 cm to represent 10 km/h on the horizontal axis and 2 cm to represent 1 litre of petrol

on the vertical axis, plot the graph of C against S for 8010 S . [3] (c) At what speed is it most economical for a journey? [2] (d) At one stage of the journey, the vehicle maintained a constant speed of 55 km/h for 45

kilometres. From the graph, estimate the number of litres of petrol that were used for this journey. [2]

(e) Between what speeds must the car be driven so that the petrol consumption do not exceed 6

litres for every 100 kilometres driven? [2]

38

92A

B

C

D E

X

6

6 Figure 1 below shows the cumulative frequency curve which represents the timings of 200 students who participated in a 7 km run.

(a) The grouped frequency table for the timings of their 7 km run is shown below. Find

the values of p and q. [2] (b) Using the grouped frequency table, calculate an estimate of (i) the mean timing, [2] (ii) the standard deviation. [2] (c) A student will pass the 7 km run if he can complete within 50 minutes. Find the

percentage of students who failed. [2] (d) Using the graph, find the median timing. [1] (e) The timings of another group of 200 students have the same median but a smaller

standard deviation. Describe how the cumulative frequency curve will differ from the curve in Figure 1. [1]

Time (x mins) 0 x < 20 20 x < 40 40 x < 60 60 x < 80 80 x < 100

Frequency 16 64 p 20 q

Time / mins Figure 1

No.

of s

tude

nts

40

80

120

160

200

100 80 6040 20 0 0

7

7 (a) The sequence 4, 5, 7, 11, , is generated by the formula baUU nn +=+1 and 41 =U . Find the values of a and b. [3]

(b) A second sequence is defined by nn VV 21

1 =+ and 160 =V .

(i) Write down the value of 1V . [1] (ii) Write nV in terms of 0V . [1] (iii) Hence, or otherwise, write down the value of nV as n becomes very large. [1]

(c) A bag contains five counters, one marked with letter A, one with letter C, one with letter S, two

with letter I. The counters are drawn at random from the bag, one at a time, without replacement. Calculate the probability that

(i) the first two counters to be drawn out will each have the letter I marked on them, [1] (ii) the second counter to be drawn out will have the letter A marked on it, [2] (iii) the order in which the counters are drawn will spell out the word ACSI. [2]

8

8 In the diagram below, OPQ is a triangle. S is a point on PQ such that PS: SQ = 2 : 1. The side OQ is produced to the point R such that OQ : QR = 3 : 2. It is given that OP = p, OQ = q.

(a) Find, in terms of p and/or q, and simplify

(i) PQ , [1]

(ii) PS , [1]

(iii) OS , [1]

(iv) OR . [1]

(b) Show that pq31=SR . [2]

(c) Given that T is the point on OP such that p95=OT , express, as simply as possible, in terms

of p and q, the vectorTR . [1]

(d) Show that SRcTR = , where c is a constant. Hence, write down two facts about TR and SR. [3]

(e) Calculate ORTPTS

ofareaofarea . [2]

T

R

P

S

O

Q

9

9 Two ships A and B are on the bearings of 029 and 122 from a port P respectively. The distance

between A and P is 700 m. A vertical tower, PT, stands at P. If the angles of depression from the top of

the tower to A and B are 25 and 12 respectively, show

(a) that the height of the tower is approximately 326 m. [2] Hence, calculate (b) the distance PB, giving your answer to the nearest m, [2] (c) the distance AB, giving your answer to the nearest m, [3] (d) the bearing of A from B, [3] (e) how far B is due east of A, giving your answer to the nearest m. [2]

700 m

B

A P

T

29

122

N

10

10 A cone of height 16 cm is made such that it just fits over a ball with radius 6 cm. Diagram I shows the

vertical cross-section of the cone and the ball with its centre at O. AB, BC and AC are tangents to the

circle.

(a) Prove that triangle OAT is similar to triangle CAM. [3] (b) Find the length of AT. Hence prove that the radius of the circular base of the cone is 12 cm. [4]

(c) CalculateABCOAT

ofareaofarea . [2]

(d) Find the curved surface area of the cone. Give your answer correct to the nearest cm2. [2] The cone is cut along its sloping edge and laid flat to form the sector ABCB of a circle of radius AB as

shown in Diagram II, find

(e) the reflex angle BAB. [2]

End of Paper 2

6 cm

A

T

O

B M C

A

C

B B

16 cm

Diagram I Diagram II

11

Answer Key 1a) x = 3 b) $300 c) $132 d) 1.20

2a) $(7.50-x) b) x5.7

54

c) showing d) $3

3a)

403020154520

b)

705638358045

c) Each element shows the number of Dark Chocolate and Red Velvet cupcakes sold in their three different sizes for the first two weeks of July.

d)

138121

e) Each element shows the average amount received from the sales of the cupcakes in the first two weeks of July.

f) (518). It shows the total amount received from the sales of the cupcakes in the first two weeks of July.

4a) ADBtosimilarisACE

quadcylicofextDBACEAcommonDABCAE

=

=.)(

)(

b) 88 c) 126 d) 82 5a) 5.8 c) 28 km/h d) 2.43 litres e) 3.620.14 S

6a) p = 92, q = 8 bi) 44 bii) 18 c) 32% d) 44 e) The graph will be steeper in the middle.

7a) a = 2, b = 3 bi) 8 bii) 0)21( VV nn = biii) 0

ci) 101 cii)

51 ciii)

601

8ai) pqPQ = aii) )(32 pqPS = aiii) )2(

31 qpOS += aiv) qOR

35=

b) showing c) )3(95 pqTR =

d)

SRTR

colinearareRandSTspoandparallelareSRandTR

SRTR

35

.,int35

=

= e)

258

9a) 326 m b) 1534 m c) 1719 m d) 326 e) 962 m

10a)

CAMtosimilarisOATcommonCAMOAT

radiustolarperpendicuTangentCMAOTAradiustolarperpendicuTangent

=

==)(

)(90 b) AT = 8 cm c)

81 d) 754 cm2

e) 216