Tampines Prelim 2009 Em 2

TAMPINES SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS

MATHEMATICS PAPER 2 2 September 2009 Additional materials:

4016 / 2 2 hours 30 minutes 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 place in the case of angles in degrees, unless a different level of accuracy is specified in the question. 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 part question. The total of the marks for this paper is 100. Indicate the calculator model used on the top right-hand corner of the first page of the answer sheets.

Setter:

Mdm Shareena Md Saniff August 2009

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

Mathematical Formulae Compound Interest r   Total amount = P1    100 

n

Mensuration Curved surface area of a cone = rl Surface area of a sphere = 4r 2 1 2 r h 3

Volume of a cone =

Volume of a sphere = Area of a triangle ABC =

4 3 r 3 1 absin C 2

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

1 2 r  , where  is in radians 2

Trigonometry

a b c   sin A sin B sin C

a 2  b 2  c 2  2bc cos A

Statistics Mean =

Standard deviation =

fx f

fx 2  fx     f  f 

Page 2

2

4016/2 Answer all the questions

1.

x2  9y2

[1]

(ii) x 2  6 xy  9 y 2

[1]

(a) Factorise (i)







Hence or otherwise factorise 5 x 2  9 y 2 – 2 x 2  6 xy  9 y 2



[2]

(b) Express as a single fraction in its simplest form, 4 1 .  2 x  5 x  4x  5

[2]

(c) (i) Express x2 – 8x + 20 in the form  x  a   b .

[1]

(ii) Hence solve the equation x2 – 8x + 20 = 5.

[2]

2

P 2. 63o 15 12 T

72 o

Q `

S

R 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]

Page 3

4016/2 3.

(a) A shopkeeper buys 15 kg of type A coffee powder at $x per kg, and 25 kg of type B coffee powder at $y per kg. He mixes the two types of coffee powder and packs the mixture into packets each of which contains 100 g of 3  1 the mixture. He sells the packets for $  x  y  each. 40   16 (i) Write down in terms of x and y, an expression for (a) the amount of money he spent on the coffee powder, (b)

the total amount of money he received for selling all the packets of coffee powder.

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

(b) A sports shop sells two types of tennis rackets, Premier and Grande. The 2 price of a Grande racket is that of a Premier racket. 3 (i) In a particular month, the shop received $3 915 in selling 60 tennis rackets. Given that 15 of the rackets sold are Premier rackets, show that the selling price of a Grande racket is $58.

[1]

[2]

[1]

[2]

(ii) If the shopkeeper made a 20% profit on each Premier racket sold and a 16% profit on each Grande racket sold, find the percentage profit made from selling the 60 tennis rackets. [3]

4

The first four terms in a sequence of numbers p 1, p2, p3, p 4. …, 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 p 6 = 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 p n = 92, find n.

[2]

Page 4

4016/2 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 to fill up the bottle. [1] (c) When the rate increases to (x + 3) cm3/s, the time taken to fill up the bottle will be reduced by 40 seconds. Write down an equation involving x and show that it can be simplified to 2x2 + 6x – 75 = 0.

[3]

(d) Solve the equation 2x2 + 6x – 75 = 0, giving your answers correct to two decimal places.

[3]

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

[2]

6.

O

In the diagram, O is the centre of the circle. AB is the diameter and it is produced 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.

Page 5

[3]

4016/2 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 possibility diagram 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]

Page 6

4016/2 8

50 12 Diagram I 32 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 nearest whole number,

[2]

(iii) the area of sector EDCM,

[2]

(iv) the volume of the chest.

[2]

2 20 3 Diagram II

(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.

Page 7

[2]

4016/2

9.

Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation 18 y  2x  2 . x 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 1  x  6 . Using a scale of 1 cm to 1 unit, draw a vertical y-axis for 6  y  22 . On your axes, plot the points given in the table and join them with a smooth curve.

[3]

(c)

Use your graph to find two solutions of 2 x 

18  10 in the range x2

1 x  6.

[2]

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 1  x  6 .

[1]

(f)

(i)

(d)

(ii)

Write down the x-coordinate of the points where the two graphs intersect.

[1]

Given that this value of x is a solution to the equation x3 + Ax2 + Bx + 18 = 0, find the value of A and the value of B.

[1]

Page 8

4016/2 10.

A quality control laboratory tested the lifespans of light bulbs. The batch of 50 light bulbs was tested. The cumulative frequency curve below shows the result.

Cumulative frequency

Lifespan of light bulbs (months) (a)

Use the diagram to find, for the lifespan of the light bulbs, (i) the median, (ii)

(b)

(c)

the 12th percentile.

(e)

[1]

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]

Copy and complete the grouped data frequency table of the lifespan of the light bulbs.

[2]

Lifespan (x months) Frequency (d)

[1]

30  x  40 40  x  50 50  x  60

60  x  70 70  x  80

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

[2]

(ii)

[2]

the standard deviation.

If two light bulbs are selected at random from these 50 light bulbs, find the probability that both of the light bulbs will last more than 60 months. Page 9

[2]