2012_prelim_iii__maths_4016-2

CHS 2012 PRELIM III Maths 4016/2 Page 1

CATHOLIC HIGH SCHOOL

PRELIMINARY EXAMINATION III

Subject : Mathematics 4016/2 Date : 13 September 2012

Level : : Secondary 4 Time : 11 00 13 30

Name : _________________________ ( ) Class : Sec 4 - ____

This document consists of 10 printed pages.

READ THESE INSTRUCTIONS FIRST

Write your name, class and index number in the spaces at the top of this page.

Write in dark blue or black pen. You may use a pencil for any diagrams or graphs.

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. 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.


Mathematical Formulae

Compound interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = lr

Surface area of a sphere = 2 4 r

Volume of a cone = hr 3

1 2

Volume of a sphere = 3

3

4r

Area of triangle ABC = Cba sin 2

1

Arc length = r , where is in radians

Sector area = 2 2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sin

sin

sin

Acbcba cos 2 222

Statistics

Mean =

f

xf

Standard deviation =

22

f

xf

f

xf


1 (a) (i) Factorise fully 221 5 6x x [1]

(ii) Express as a single fraction in its simplest form

2

4 7 4

21 5 6 2 3

x

x x x

[3]

(b) Given 5

23

nm k

n

, express n in terms of k and m. [3]

(c) (i) Expand

21

xx . [1]

(ii) Hence find the value of 2

2 1

xx , given that

18x

x . [2]

2 In 2011, Victor earned a monthly salary of $3500 and received a 13th

month bonus and a

Performance Bonus equivalent to 1.25 months.

In his country, income tax is payable by all employees who earns more than $25 000 per

annum in the preceding year. Tax payable is 4.25% of the amount exceeding $25 000.

The total amount of tax deductible which he can claim as reliefs is $2866.

(a) Calculate his annual income earned in 2011. [2]

(b) Calculate his income tax payable for 2012. [2]

He has opted not to pay his income tax as a lump sum but to take advantage of the interest-

free option of spreading out the payment over 12 months.

(c) Calculate the tax he has to pay each month for the next 12 months. [1]

Victor has painstakingly saved $10000 for 2011, and is considering two saving plans as

follows:

He plans to take either plan for a period of 10 years.

(d) Calculate the interest he will earn in 10 years under Plan B. [2]

(e) Which Plan will give him a better return? Justify your answer. [2]

Plan A: Simple interest of 4.5% per annum

Plan B: 4% per annum compounded every three months


3

In the diagram, P, Q, R and S lie on the bigger circle in which 58PRQ and 32PQS .

R, S, T and U lie one the smaller circle, centre O, in which 120RUT and TU = RU.

It is given that URQ is a straight line.

(a) Find

(i) PRS , [1]

(ii) TSR , [1]

(iii) ROU . [2]

(b) Explain why QS is a diameter of the bigger circle. [1]

(c) Show that

(i) RTS is an equilateral triangle, [3]

(ii) U, O and S are collinear. [2]

O

32

T

U

Q

R

S

P

120 58


4

The diagram shows three circles of radius 5cm, 2 cm and x cm, touching each other.

The three lines joining the centres of the circles form a right-angled triangle as shown.

(a) From the information given above, form an equation and show that it reduces to 2 7 10 0x x . [3]

(b) Solve the equation 2 7 10 0x x , giving your solutions correct to 2 decimal

places. [3]

(c) Find the radius of the circle that passes through the centres of these circles. [1]

5 The first four terms in a sequence of numbers, T1, T2, T3, T4, .. are given below.

T1 = 0 23 1 2 6

T2 = 1 23 4 3 16

T3 = 2 23 7 4 32

T4 = 3 23 10 5 62

(a) Write down an expression for T5 and show that T5 = 130. [1]

(b) Write down the value for T6. [1]

(c) Find an expression, in terms of n, for the nth term, Tn, of the sequence. [3]

(d) Show that Tn+1 Tn = 12 3 3n n . [3]


6

In the diagram, ASB is a semi-circle with AB as the diameter. ACB is an arc of a circle,

centre O, radius 5 cm, and 2.4AOB radians.

Calculate

(a) the length of the diameter AB, [2]

(b) the area of the segment ACB, [3]

(c) the area of the shaded region. [2]

7

A ship leaves a port at P and sails 21 km

towards a lighthouse L. It then sails 28 km towards

an island I. The bearing of L from I is 116 and the bearing of P from I is 163.

(a) Calculate

(i) ILP , [4] (ii) the distance IP, [2] (iii) the bearing of P from L. [2]

(b) The ship then returns to the port P by travelling along the route IP. Calculate the

distance left on that journey when the ship is closest to the lighthouse L. [2]

(c) Given that the height of the lighthouse is 500 metres, calculate the angle of depression of point P when viewed from the top of the lighthouse. [2]

North

P

L

I

28

21

2.4 O

A

S C

B

5


8 A company manufactures ping- pong balls, each of diameter 4 cm.

Diagram I shows a cross-section of 7 ping-pong balls packed in a rectangular box.

(a) Find the volume of each ping-pong ball. [1]

(b) Calculate the total surface area of the rectangular box. [2]

(c) Calculate the amount of space within the box which is not occupied. [3]

The company wants to change the packaging. The manager suggests a regular hexagonal

box.

Diagram II shows the plan and side view of the new box.

(d) Find the length of one side of the box. [3]

(e) Which box, rectangular or hexagonal, will cost less if the same material is used

in the making of the box. Justify your answer. [3]

Diagram I

Diagram II


9

In the diagram, ,OA OB a b and 5

9OC b . D and E are points on BA such that

BD : DA = 1 : 2 and BE : EA = 3 : 1.

(a) Express, as simply as possible, in terms of a and/or b,

(i) BA , [1]

(ii) BD , [1]

(iii) BE , [1]

(iv) OE , [1]

(v) CD . [1]

(b) Hence show that CD is parallel to OE. [1]

(c) Find the ratio of

(i) BOE

BCD

of area

of area, [1]

(ii) area of

area of

BCD

BOA

. [1]

B

D

C

b

O A a

E


10 The diagram below shows the cumulative frequency curve for the marks of 400 pupils

who sat for a Mathematics examination.

(a) Use the graph to find

(i) the median mark, [1]

(ii) the interquartile range, [1] (iii) the passing mark for the examination if 45% of the students passed the

examination. [1]

(b) The same 400 students also sat for a Science examination. The box and whisker

diagram below illustrates the marks obtained.

64 27 2925

Examination Mark


Use the diagram to find

(i) the median marks, [1]

(ii) the interquartile range. [1]

(c) Compare the marks obtained by the 400 students for the Mathematics and Science

examinations in two different ways. [2]

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

The variables x and y are connected by the equation 2 4

5 .10

xy

x

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

x 0.5 0.7 1 2 3 4 5 6 7 8

y 3.0 0.8 0.9 2.6 2.8 2.4 1.7 0.7 k 1.9

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

(b) Taking 2cm to represent 1 unit on each axis, draw x and y axes for 0 8x and

3 3y .

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

(c) By drawing a tangent, find the gradient of the curve at the point when x = 2. [2]

(d) Use your graph to find

(i) the x-coordinate of the point on the curve at which the tangent is parallel to

the line 8y x , [2]

(ii) the values of x in 0.5 8x for which 3 22 40 40 0x x x . [2]

2012_prelim_iii__maths_4016-2

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