INNOVA JUNIOR COLLEGE

PRELIMINARY EXAMINATION 2010 H2 MATHEMATICS PAPER 1

IJC/2010/JC2 9740/01/S/10

1 For any given mass of gas, the volume V cm3 and pressure p (in suitable units) satisfy

the relationship

1 nV pk

= ,

where k and n are constants.

For a particular type of gas, 2.3n = − . At an instant when volume is 32 cm3, the

pressure 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 of 2

x in the series expansion of ( )

2

1 3n

x+ is 108, find the

value of n, where n is 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) Find 1

n

r

r

u=∑ in terms of n. [2]

IJC/2010/JC2 9740/01/S/10 [Turn over

3

4

The diagram shows a quadrilateral OABC with OA AB= and OC BC= .

Points A and B have position vectors

1

2

α

and

1

0

1

−

respectively.

(a) Find cosine of angle OAB in terms of α . [2]

(b) M is the midpoint of OB and 4AM MC=uuuur uuuur

. By considering the area of the

quadrilateral OABC, show that

5

2OA OC OA OB× = ×uuur uuur uuur uuur

. [4]

5 Illustrate, on a single Argand diagram, the locus of the point representing the complex

number z that satisfies both the inequalities

( )3

arg 3 3i4 2

zπ π

− < − − ≤ and 3 3i 2z − − ≤ . [4]

(i) Find the greatest and least values of 3iz + . [2]

(ii) Find the least possible value of ( )zarg , giving your answer in radians. [3]

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

d

d3

w

xw= − . [2]

Hence find y in terms of x, given that 1

2y = − when 2x = . [4]

C

B

O

A

M

INNOVA JUNIOR COLLEGE

PRELIMINARY EXAMINATION 2010 H2 MATHEMATICS PAPER 1

IJC/2010/JC2 9740/01/S/10

7 A curve is defined parametrically by

2 cosx t= , 2 1y t= − ,

where 0 t π< ≤ .

(i) Find the equation of the normal to the curve at the point P where 3

tπ

= . [5]

(ii) The normal at P meets the y-axis at N and the x-axis at M. Given that the curve

meets the y-axis at Q, find the area of triangle MNQ, correct to 1 decimal place.

[5]

8 A curve has equation 2 2( 4)

14 9

x y −− = .

(i) Sketch the curve, stating the equations of the asymptotes and the coordinates

of the vertices. [3]

(ii) The region enclosed by the curve 2 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 of

revolution of volume V. Find the exact value of V, giving your answer in terms

of π . [4]

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 puts y insects at the beginning of the first week and then a further y 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 of y. [1]

(ii) Show that, at the end of n weeks, 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 insects

to exceed 12513 −y . [4]

IJC/2010/JC2 9740/01/S/10 [Turn over

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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 − on

the same diagram. [3]

(iii) Find the exact solutions of the equation

1f ( ) 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 for

0x > . [2]

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 dx x x∫ , [2]

(ii) the exact value of 4

0cos3 cos dx x x x∫

π

. [4]

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

the graph of 3

sin 22

y x π

= −

. [2]

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

cos3 cosy x x= and 3

sin 22

y x π

= −

for 02

xπ

≤ ≤ . [3]

INNOVA JUNIOR COLLEGE

PRELIMINARY EXAMINATION 2010 H2 MATHEMATICS PAPER 1

IJC/2010/JC2 9740/01/S/10

12 A plane 1π has equation

2

1 9

3

− =

r. and a line 1l has equation

3 4

0 1

0 2

= + λ −

r .

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

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

[2]

The equations of planes 2π and 3π are given as

2π :

1 2 2

1 0 2

1 3 1

s t

= + + −

r , and

3π : x y zα β+ − = , where ,α β ∈ � .

(ii) Find the equation of plane 2π in the form d�r n = . Explain why the planes

1π and 2π intersect. [4]

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

3 2

3 1

2 1

+ µ −

r = .

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

find the conditions satisfied by α and β . [3]