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INNOVA JUNIOR COLLEGEJC 2 PRELIMINARY EXAMINATION 2in preparation for General Certificate of Education Advanced Level

Higher 2

CANDIDATENAME

CLASS INDEX NUMBER

MATHEMATICS

Paper 1

Additional Materials: Answer Paper

Graph PaperList of Formulae (MF15)

9740/01

15 September 2010

3 hours

READ THESE INSTRUCTIONS FIRST

Do not open this booklet unti l you are told to do so.

Write your name, class and index 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 soft pencil for anydiagrams or graphs.

Do not use staples, paper clips, highlighters, glue or correction flu id.

Answer allthe questions.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in thecase of angles in degrees, unless a different level of accuracy is specified in the question.

You are expected to use a graphic calculator.

Unsupported answers from a graphic calculator are allowed unless a question specificallystates otherwise.Where unsupported answers from a graphic calculator are not allowed in a question, youare required to present the mathematical steps using mathematical notations and notcalculator commands.

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.

This document consists of 6printed pages.

Innova Junior College [Turn over

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1 For any given mass of gas, the volume Vcm3and pressurep(in suitable units) satisfy

the relationship1 nV pk

,

where kand nare constants.

For a particular type of gas, 2.3n . At an instant when volume is 32 cm3, thepressure is 105 units and the pressure is increasing at a rate of 0.2 units s

1.

Calculate the rate of decrease of volume at this instant. [4]

2 Given that the coefficient of2x in the series expansion of

2

1 3 n

xis 108, find the

value of n, where nis a positive integer. [4]

3 The sequence of numbers nu , where 1, 2, 3,...,n is such that

1

9

8u and 18 7 8n nu u n .

Use the method of mathematical induction to show that

32 nnu n for 1n . [5]

(i) Determine if the sequence converges. [1]

(ii) Find1

n

r

r

u

in terms of n. [2]

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6 Show, by means of the substitution 2w x y , that the differential equation

d

d2 3 0

y

xx y xy

can be reduced to the form

dd

3wx

w . [2]

Hence findyin terms ofx, given that1

2y when 2x . [4]

7 A curve is defined parametrically by

2cosx t , 2 1y t ,

where 0 t .

(i) Find the equation of the normal to the curve at the point Pwhere3

t . [5]

(ii) The normal at Pmeets they-axis atNand thex-axis atM. Given that the curve

meets they-axis at Q, find the area of triangleMNQ,correct to 1 decimal place.

[5]

8 A curve has equation2 2( 4)

14 9

x y .

(i) Sketch the curve, stating the equations of the asymptotes and the coordinatesof the vertices. [3]

(ii) The region enclosed by the curve2 2( 4)

14 9

x y , the x-axis and the line

2x is rotated through 4 right angles about the y-axis to form a solid ofrevolution of volume V. Find the exact value of V, giving your answer in terms

of . [4]

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9 In a medical research centre, a particular species of insect is grown for treatment of

open wounds. The insects are grown in a dry and cool container, and they are left to

multiply. The increase in the number of insects at the end of each week is at a

constant rate of 4% of the number at the beginning of that week. At the end of each

week, 10 of the insects would die due to space constraint and are removed from the

container.

A researcher putsy insects at the beginning of the first week and then a furthery at the

beginning of the second and each subsequent week. He also decides that he will not

take any live insects out of the container.

(i) How many insects will there be in the container at the end of the first week?

Leave your answer in terms ofy. [1]

(ii) Show that, at the end of nweeks, the total number of insects in the container is

26 250 1.04 1n

y

. [4]

(iii) Find the minimum number of complete weeks for the population of the insectsto exceed 12513 y . [4]

10 The functions f and g are defined as

f : 1x x for 1x

g : e 1xx for 0x

(i) Define 1f in a similar form, including its domain. [3]

(ii) State the relationship between f and 1f , and sketch the graphs of f and 1f onthe same diagram. [3]

(iii) Find the exact solutions of the equation1f ( ) f ( )x x . [2]

(iv) Show that the composite function fg exists. [2]

(v) Given that h ( ) fg( )x x for 0x , show that h is an increasing function for0x . [2]

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11 (a) Write 2cos3 cosx x in the form cos cospx qx , where p and q are

positive integers. [1]

Hence find

(i) cos3 cos d x x x , [2]

(ii) the exact value of 40

cos3 cos d x x x x

. [4]

(b) State a sequence of transformations which transform the graph of siny x to

the graph of3

sin 22

y x

. [2]

(c) Find the numerical value of the area of the region bounded by the curves

cos3 cosy x x and3

sin 22

y x

for 02

x

. [3]

12 A plane 1 has equation

2

1 9

3

r. and a line 1 has equation

3 4

0 1

0 2

r .

(i) Find the coordinates of P, the point of intersection of 1 and 1 . [4]

Hence, or otherwise, find the shortest distance from point A (3, 0, 0) to 1 .[2]

The equations of planes 2 and 3 are given as

2 :

1 2 2

1 0 2

1 3 1

s t

r , and

3 : x y z , where , .

(ii) Find the equation of plane 2 in the form dr n = . Explain why the planes

1 and 2 intersect. [4]

(iii) The line of intersection of planes 1 and 2 is 2 . The line 2 has equation

3 23 1

2 1

r = .

Given that the three planes 1 , 2 and 3 do not have any points in common,

find the conditions satisfied by and . [3]