ANGLO-CHINESE JUNIOR COLLEGE

MATHEMATICS DEPARTMENT

MATHEMATICS
Higher 2 9740
Paper 1
14 August 2008
JC 2 PRELIMINARY EXAMINATION
Time allowed: 3 hours

Additional Materials: List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your Index number, Form Class, graphic and/or scientific calculator model/s on the cover page.
Write your Index number and full name on all the work you hand in.
Write in dark blue or black pen on your answer scripts.
You may use a soft pencil for any diagrams or graphs.
Do not use paper clips, highlighters, glue or correction fluid.

Answer all the questions.

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.
You are expected to use a graphic calculator.
Unsupported answers from a graphic calculator are allowed unless a question specifically states
otherwise.
Where unsupported answers from a graphic calculator are not allowed in the question, you are
required to present the mathematical steps using mathematical notations and not calculator commands.
You are reminded of the need for clear presentation in your answers.

The number of marks is given in brackets [ ] at the end of each question or part question.
At the end of the examination, fasten all your work securely together.

This document consists of 5 printed pages.

[Turn Over

Anglo-Chinese Junior College

H2 Mathematics 9740: 2008 JC 2 Preliminary Examination Paper 1
Page 1 of 5
 ANGLO-CHINESE JUNIOR COLLEGE
MATHEMATICS DEPARTMENT
JC2 Preliminary Examination 2008

MATHEMATICS 9740
Higher 2
Paper 1
/ 100
Index No: Form Class: ___________
Name: _________________________
Calculator model: _____________________

Arrange your answers in the same numerical order.

Place this cover sheet on top of them and tie them together with the string provided.

Question no. Marks

1

5
6

10

11

12

13

14

Anglo-Chinese Junior College

H2 Mathematics 9740: 2008 JC 2 Preliminary Examination Paper 1
Page 2 of 5
1 Given that x is large such that x −2 and lower powers of x can be neglected, expand
2− x
in descending powers of x, and state the range of values for which the
1 + 3x
expansion is valid. [5]

2 Find
⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞
⎜ − + ( m − 1) d ⎟ + ⎜ − + md ⎟ + ⎜ − + ( m + 1) d ⎟ + …… + ⎜ − + ( m + n ) d ⎟ ,
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠
where m, n and d are constants, leaving your answer in terms of m, n and d. [3]

∫ e ( cos x − 1) dx .
x 2
3 Find [5]

4 An ant of negligible size walks a distance of 10 units from the origin in the x-y plane
along the x-axis. It then turns left and goes up 5 units from its current point. If the ant
continues turning left and going half the distance it had previously walked, repeating
the pattern, find the coordinates of the point where the ant will eventually end up. [4]

5 By using an algebraic method, solve the inequality x x − 2 ≥ 1 . [5]

Hence, or otherwise, find the range of values of x that satisfy ( x + 1) x − 1 ≥ 1 . [3]

π
6 The parametric equations of a curve are x = ln(cos θ ), y = ln(sin θ ), 0 < θ < . Find
2
π
the equation of the tangent to the curve at the point where θ = , leaving your
4
answer in the form of y = ax + b where a and b are exact values to be found. [4]
Explain, using an algebraic method, why the tangent will not meet the curve again. [2]

7 Newton’s Law of Cooling states that the rate of change of the surface temperature of
an object is proportional to the difference between the temperature of the object and
the temperature of its surrounding at that time.
The room temperature in a café is kept constant at 20°C by means of its central
cooling system. It was found that the temperature of a cup of coffee left on a table in
the café will drop from 60°C to 45°C in 10 minutes. Express this information as a
differential equation connecting θ in degree Celsius and t in minutes, where θ is the
temperature of the coffee at time t.
One day, Daniel bought a cup of “extra hot” coffee and noticed that it took 20 minutes
for the temperature to cool to 45°C. What is the temperature of his coffee when he first
bought it? [6]

[Turn Over
Anglo-Chinese Junior College
H2 Mathematics 9740: 2008 JC 2 Preliminary Examination Paper 1
Page 3 of 5
 ax − 1
8 A curve has the equation y = for some constants a, b. Given that the
3x + b
1
asymptotes are x = , y = 2 , find a, b and sketch the curve showing the asymptotes
3
and the coordinates of the points of intersection with the axes. [3]
ax − 1
Sketch, on a separate diagram , the curve with the equation y 2 = , showing the
3x + b
asymptotes and the coordinates of the points of intersection with the axes. [2]
1
State the sequence of transformations which transform the graph of y = to the
x
ax − 1
graph of y = . [3]
3x + b

d3y d 2 y dy
9 Given that y = sin −1 x , prove that 1 − x 2 (
dx3
− 3 )
x −
dx 2 dx
= 0 . Hence, find the

Maclaurin’s series for y up to and including the term in x5 . [7]

1
Deduce the expansion for . [2]
2
1− x
Hence estimate the value for 3 . [2]

1
10 (a) R is the region enclosed by the curve y = x 2 1 − 4 x 2 for 0 ≤ x ≤ , the
4
1 1
line x = , and the x-axis. Using the substitution x = sin θ , find the exact
4 2
area of the region R. [6]
(b) Sketch the curves y 2 = x ln x and y = 1 e x , and shade the region, A, bounded
6
by the two curves. Find the volume generated when A is rotated through 2π
radians about the x-axis. [4]

11 On an Argand diagram, sketch the set of points representing the complex number z
satisfying the equation arg(iz + 2) = 23 π . [2]
Hence find the least value of z − 3 + i . [3]
If there is exactly one complex number w that satisfies both conditions
arg(iw + 2) = 23 π and w − 3 + i = m , find the range of values of m. [2]

4a 2
12 Given that f ( x) = x + a + , x ≠ 2, 2 < a < 3 . In terms of a,
x−2
(i) find the asymptotes of y = f ( x ) . [2]
(ii) find the coordinates of the stationary points. [3]
(iii) sketch the graph of y = f ( x ) , labeling clearly the asymptotes, turning points
and axial intercepts. [3]

Anglo-Chinese Junior College

H2 Mathematics 9740: 2008 JC 2 Preliminary Examination Paper 1
Page 4 of 5
13 Write down, in the form reiθ , the roots of the equation w3 = 1 . [1]
Hence show that all the possible values of complex numbers z that satisfy the equation
kπ
( z + i)3 + ( z − i)3 = 0 can be expressed in the form z = − tan where k = −1, 0, 1. [4]
3
3 3
By expanding ( z + i) + ( z − i ) = 0 to form a polynomial with real coefficients, prove
that tan( π3 ) = 3 . [2]

14 The points A, B, C and P have position vectors given respectively by

a = 2i + j + 3k,
b = i + 3j + k,
c = 3i − 6j + 6k and
p = (2 − 2t)i + (1 + 4t)j + (3 − 4t)k.

(i) Find the obtuse angle between the line OB and the line OC. [2]
(ii) Find the area of triangle OBC. [2]
(iii) Find the values of t such that the length of the projection of CP onto the line
CB is 110 units. [4]
(iv) Show that, for all values of the parameter t, the points P, A and B are collinear.
[2]
(v) Hence, deduce the area of triangle OPC, stating your reason clearly. [2]

- End of Paper -

Anglo-Chinese Junior College

H2 Mathematics 9740: 2008 JC 2 Preliminary Examination Paper 1
Page 5 of 5