Tampines Prelim 2009 Em 2

8/14/2019 Tampines Prelim 2009 Em 2

http://slidepdf.com/reader/full/tampines-prelim-2009-em-2 1/9

This question paper consists of 9 printed pages and 1 blank page.

TAMPINES SECONDARY SCHOOL

PRELIMINARY EXAMINATION 2009

SECONDARY FOUR EXPRESS

MATHEMATICS 40

PAPER 2 2 hours 30 minutes

2 September 2009

Additional materials: Answer Paper

Graph Paper

READ THESE INSTRUCTIONS FIRST

Write your answers and working on the separate answer paper provided.Write your name, class and register number on all the work you hand in.Write in dark blue or black pen on both sides of the paper.You may use a pencil for any diagram or graph.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer all the questions.Write your answers on the papers provided.Give non-exact numerical answers correct to 3 significant figures or 1 decimal placein the case of angles in degrees, unless a different level of accuracy is specified in thequestion.The use an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers.

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 partquestion.The total of the marks for this paper is 100.

Indicate the calculator model used on the top right-hand corner of the first pageof the answer sheets.

Setter: Mdm Shareena Md Saniff

August 2009



Page 2

Mathematical Formulae

Compound Interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 24 r

Volume of a cone = hr 2

3

1

Volume of a sphere = 3

3

4r

Area of a triangle ABC = C absin2

1

Arc length = r , where is in radians

Sector area = 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

Abccba cos2222

Statistics

Mean = f

fx

Standard deviation =

22

f

fx

f

fx



4016/2

Page 3

Answer all the questions

1. (a) Factorise (i) 22 9 y x [1]

(ii) 22 96 y xy x [1]

Hence or otherwise factorise 5 22 9 y x – 2 22 96 y xy x [2]

(b) Express as a single fraction in its simplest form, [2]

54

1

5

42

x x x

.

(c) (i) Express x2 – 8 x + 20 in the form ba x 2

. [1]

(ii) Hence solve the equation x

2

– 8 x + 20 = 5. [2]

2.

`

In the diagram, RST is a straight line. Angle PST = 90 , angle QPS = 63 ,angle PSQ = 72 , PS = 12 m and PT = 15 m. Calculate

(a) angle PTS, [2]

(b) PR, [2]

(c) QS, [2]

(d) the area of triangle PQS. [2]

(e) A flag pole, 25 m tall, is placed vertically upright at Q.Find the angle of elevation of the top of the flag pole from S. [2]

63o

72

o

P

Q

R

S

T

15

12



4016/2

Page 4

3. (a) A shopkeeper buys 15 kg of type A coffee powder at $ x per kg, and 25 kgof type B coffee powder at $ y per kg. He mixes the two types of coffeepowder and packs the mixture into packets each of which contains 100 g of

the mixture. He sells the packets for $

y x

40

3

16

1each.

(i) Write down in terms of x and y, an expression for(a) the amount of money he spent on the coffee powder, [1]

(b) the total amount of money he received for selling all the packetsof coffee powder. [2]

(ii) Find the profit made, in terms of x and y, giving your answer as simplyas possible. [1]

(b) A sports shop sells two types of tennis rackets, Premier and Grande. The

price of a Grande racket is

3

2that of a Premier racket.

(i) In a particular month, the shop received $3 915 in selling 60 tennisrackets. Given that 15 of the rackets sold are Premier rackets, showthat the selling price of a Grande racket is $58. [2]

(ii) If the shopkeeper made a 20% profit on each Premier racket sold and a16% profit on each Grande racket sold, find the percentage profit madefrom selling the 60 tennis rackets. [3]

4 The first four terms in a sequence of numbers p1, p2, p3, p4. …, are given below.

p1 = 5 1 – 2 0 = 5

p2 = 5 2 – 2 1 = 8

p3 = 5 3 – 2 2 = 11

p4 = 5 4 – 2 3 = 14

(a) Write down the expression for p6 and show that p6 = 20. [1]

(b) Write down an expression for p7 and evaluate it. [1]

(c) Find an expression, in terms of n, for the nth term, pn, of the sequence.Leave the expression in its simplest form. [2]

(d) Evaluate p17. [2]

(e) If the value of pn = 92, find n. [2]



4016/2

Page 5

5. A bottle has a capacity of 500 cm3. It is filled with water at a rate of x cm3 /s.

(a) Express, in terms of x, the time taken to fill up the bottle. [1]

(b) If the rate increases to ( x + 3) cm3 /s, express, in terms of x, the time taken tofill up the bottle. [1]

(c) When the rate increases to ( x + 3) cm3 /s, the time taken to fill up the bottlewill be reduced by 40 seconds. Write down an equation involving x andshow that it can be simplified to 2 x2 + 6 x – 75 = 0. [3]

(d) Solve the equation 2 x2 + 6 x – 75 = 0, giving your answers correct to twodecimal places. [3]

(e) Hence find the original time taken, in seconds, to fill up the bottle. [2]

6.

In the diagram, O is the centre of the circle. AB is the diameter and it isproduced at T . CT is a tangent to the circle at C and angle BAC = 34 .

(a) State the reason why angle OCT = 90 . [1]

(b) Find(i) angle ATC , [2]

(ii) angle BCT . [2]

(c) Show that triangles ACT and CBT are similar. [3]

O



4016/2

Page 6

7. Box A contains 6 pieces of paper numbered 2, 3, 4, 5, 6 and 7.Box B contains 5 pieces of paper numbered 1, 3, 5, 7, and 9.One piece of paper is removed at random from Box A and then Box B.

(a) Find the probability that the two numbers obtained have(i) the same value, [2]

(ii) a product that is exactly divisible by 6. [2]

(b) The sums of the two numbers obtained can be represented in a possibilitydiagram below.

+ 2 3 4 5 6 7

1

3

5

7

9

(i) Copy and complete the possibility diagram. [2]

(ii) Using the diagram, or otherwise, find the probability that the sum of the two numbers is(a) even, [1]

(b) a prime number, [1]

(c) greater than or equal to 9. [1]



4016/2

Page 7

8

Diagram I shows a chest which has a uniform cross-section ABCDE , in which ABCE

is a rectangle. M is the midpoint of AB. CDE is an arc of a circle with M as the centre. AB = 32 cm, BC = 12 cm and BQ = 50 cm.

(a) Show that angle CME = 1.85 radians, [2]

(b) Find(i) the length of the arc EDC , [3]

(ii) the surface area of the lid, CDETSR, leaving your answer to the nearestwhole number, [2]

(iii) the area of sector EDCM , [2]

(iv) the volume of the chest. [2]

(c) Diagram II shows a bar of chocolate with dimensions 3 cm by 20 cm by 2 cm.They are being kept in the chest as shown above.Find the number of bars of chocolates that can fit into the chest. [2]

20

3

2

Diagram II

32

Diagram I

12

50



4016/2

Page 8

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

The variables x and y are connected by the equation

2

182

x x y .

Some corresponding values of x and y are given in the following table.

x 1 1.5 2 3 4 5 6

y 20 11 8.5 8 9.1 10.7 q

(a) Find the value of q. [1]

(b) Using a scale of 2 cm to 1 unit, draw a horizontal x-axis for 61 x .

Using a scale of 1 cm to 1 unit, draw a vertical y-axis for 226 y .

On your axes, plot the points given in the table and join them with asmooth curve. [3]

(c) Use your graph to find two solutions of 1018

22

x x in the range

61 x . [2]

(d) By drawing a tangent, find the gradient of the graph at the point where x = 3.5. [3]

(e) On the same axes, draw the graph of y = x + 10 for 61 x . [1]

(f) (i) Write down the x-coordinate of the points where the two graphsintersect. [1]

(ii) Given that this value of x is a solution to the equation x

3+ Ax

2+ Bx + 18 = 0, find the value of A and the value of B. [1]



4016/2

Page 9

10. A quality control laboratory tested the lifespans of light bulbs. The batchof 50 light bulbs was tested. The cumulative frequency curve belowshows the result.

(a) Use the diagram to find, for the lifespan of the light bulbs,(i) the median, [1]

(ii) the 12th

percentile. [1]

(b) Light bulbs that could not last for at least 72 months will be destroyed.Calculate the percentage of light bulbs that will have to be destroyed. [2]

(c) Copy and complete the grouped data frequency table of the lifespan of thelight bulbs. [2]

Lifespan

( x months)

4030 x 5040 x 6050 x 7060 x 8070 x

Frequency

(d) Using your grouped frequency table, calculate an estimate of (i) the mean lifespan of the light bulbs, [2]

(ii) the standard deviation. [2]

(e) If two light bulbs are selected at random from these 50 light bulbs, findthe probability that both of the light bulbs will last more than 60 months. [2]

Lifespan of light bulbs (months)

Cumulativefrequency

Documents

Tampines Prelim 2009 Em 2