S.7 Mock Exam. 09-10 / Pure Maths. 2 / p.1
2010-AL
Pure Mathematics
Paper 2
LIU PO SHAN MEMORIAL COLLEGE
(2009-2010)
MOCK EXAMINATION
SECONDARY SEVEN
PURE MATHEMATICS PAPER 2
Date : 9 - 2 – 2010
Time Allowed : 3 hours
1.
This paper consists of Section A and Section B.
2.
Answer ALL questions in Section A.
3.
Answer any FOUR questions in Section B.
4.
Unless otherwise specified, all working must be clearly shown.
FORMULAS FOR REFFERENCE
sin  A  B = sin A cos B  cos A sin B
sin A + sin B = 2 sin
A+B
A-B
cos
2
2
sin A - sin B = 2 cos
A+B
A-B
sin
2
2
cos  A  B = cos A cos B  sin A sin B
tan  A  B =
tan A  tan B
1  tan A tan B
cos A + cos B = 2 cos
A+B
A-B
cos
2
2
cos A - cos B = -2 sin
A+B
A-B
sin
2
2
2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
2 sin A sin B = cos (A - B) - cos (A + B)
S.7 Mock Exam. 09-10 / Pure Maths. 2 / p.2
Section A (40 %)
Answer ALL questions in this section.
1.
(a) Evaluate
(b)
lim
x0
1
 2x  8x

2

x
 .

Let k be a real constant and f : R  R be defined by
 k
1
f x   
 1 - sin x  x
when x  0
when x  0
.
It is given that f (x) is continuous at x = 0.
Find the value of k.
(6 marks)
2.
L et
f : R  R be a function satisfying
(i)
f (x + y) = f (x) cos y + f (y) cos x
for all x, y R.
(ii) f is differentiable at x = 0 with f ’(0) = 1.
(a) Find f (0).
(b) Show that f is differentiable for all x  R with f ’(x) = cos x.
Hence find f (x).
(6 marks)
3.
Let
f : R  R be a function defined as
f (x) =
lim
n 
x 2n  1 - x
1  x 2n
for all x  R
where n is a positive integer.
(a) Show that f is an odd function.
(b) Show that for all x >1,
f (x) = x.
(c) Sketch the graph of y = f (x).
(d) Is f an injective function? Explain your answer.
(7 marks)
4.
(a) By using the substitution t = tan


, evaluate
2
1
d .
5  4 cos 
(b) Hence, or otherwise, show that
1 n
lim
n
4
1
1
tan -1 .
=
n k =1
3
3
  k 
5  4 c o s - 

n 
 2

(7 marks)
S.7 Mock Exam. 09-10 / Pure Maths. 2 / p.3
5.
(a) Prove that the line y = mx + c touches the conic  x 2   y 2  1 , where   0,
if and only if
c2 
   m2
.

(b) Find the equations of the common tangents to the ellipse x 2  2 y 2  18 and the hyperbola
7 x 2  38y 2  266 .
(7 marks)
6.
(a) Evaluate

cos 2  d
and

cos 3  d .
(b) Let P be a fixed point on a circle with radius 2. When the circle rolls along the positive x-axis
without slipping, the locus of P is called a cycloid.
As shown in the above figure, the circle starts rolling with P at the origin and reaches the position
after the circle has rolled through an angle  , where 0    2 .
(i)
Show that the parametric equations of the cycloid obtained is
 x  2  sin  

 y  21  cos  
where 0    2 .
(ii) The figure below shows the cycloid for 0    2 .
Find the volume of the solid generated by revolving the region enclosed by the positive
x-axis and the cycloid about the x-axis.
(7 marks)
S.7 Mock Exam. 09-10 / Pure Maths. 2 / p.4
Section B (60 %)
Answer any FOUR questions in this section.
7.
Let
 x3 

f (x) = ln 
 x 1 


(a) Find f ’(x) and f ” (x)
where x < -1 and x > 0.
for x < -1 and x > 0.
(3 marks)
(b) Determine the range of values of x for each of the following cases :
(i)
f ’(x) > 0 ,
(ii)
f ” (x) > 0.
(3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of the graph of y = f (x).
(2 marks)
(d)
Show that the graph of y = f (x) does not have any oblique asymptote.
Find the asymptote(s) of the graph of y = f (x).
(3 marks)
(e)
Sketch the graph of y = f (x).
(2 marks)
(f)
Using the graph in (e), or otherwise, find the range of values of k such that the equation
x 3 - kx - k  0 has
(i)
exactly one real root;
(ii)
two distinct real roots;
(iii) three distinct real roots.
(2 marks)
S.7 Mock Exam. 09-10 / Pure Maths. 2 / p.5
8.
For any positive integer n, define I
(a) (i)


n
1
0
f (x) dx , where f x   1  x n e x .
Prove that f (x) is strictly decreasing for x  (0, 1) .
(ii) Show that 0  I  1 .
n
(4 marks)
(b) (i)
Show that
I
m 1
 (m  1) I
m
1
for m  1 .
(ii) Hence, or otherwise, show that
n 1
1
I  e  
n! n
k 0 k!
for n  1 .
(7 marks)
(c) Using (a) and (b), show that
n
1
= e.
k 0 k!
lim 
n
(4 marks)
9. (a)
Suppose f is continuous on [-a , a]
(i)
where a is a positive number.
Show that

0
-a

f (x) dx 
a
0
f (-x) dx .
(ii) Hence, if f (x) = f (-x) for  a  x  a , deduce that

a
-a
f (x) dx  2

a
0
f (x) dx .
(4 marks)
(b) Using the substitution t =
3
tan  , show that
3
 0 1  3t 2
1
1
dt =
3
.
9
(3 marks)
(c) Given I =
(i)

1 - t2
dt and J =
0 1  3t 2
1
Without evaluating I and J, show that

1
t2
0 1  3t 2
dt .
I + 4J = 1.
(ii) Using (b) and considering I + J, or otherwise, evaluate J.
(5 marks)
(d) Hence, or otherwise, evaluate
1 1  t2
dt .
-1 1  3t 2

(3 marks)
S.7 Mock Exam. 09-10 / Pure Maths. 2 / p.6
10. Given a rectangular hyperbola H: xy  c
2
where c  0.
c
(a) Find the equation of the normal to H at P(ct, ) where t  0.
t
(3 marks)
c
(b) It is given that the normal to H at P cuts the hyperbola H again at the point Q(cs, ) where s  0.
s
(i)
Show that t 3 s  1 .
(ii) Hence, or otherwise, prove that the there is one and only one straight line which is normal to
the hyperbola H at both points of intersection.
(6 marks)
(c) It is given that the normal to H at P cuts the x-axis at R and y-axis at S.
(i)
Prove that the mid-point of RS is also the mid-point of the normal chord PQ.
(ii) Hence, or otherwise, show that the locus of the mid-point of normal chords of the hyperbola H
is given by the equation 4x 3 y 3  c 2 (x 2  y 2 ) 2  0 .
(6 marks)
11. (a) Let f (x) and g (x) be two continuous functions defined on [a , b] and f ’(x)  0 and g (x)  0
for all x  [a , b].
(i)
For all t  [a , b], show that
f (b)  f (t)  f (a).
(ii) Define F (x) =

b
a
f (t) g (t) dt - f (x)

b
a
g (t) dt , show that F(a)  0 and F (b)  0 .
(iii) Hence, or otherwise, show that there exists c  [a , b] such that

b
a
f (t) g (t) dt = f (c)

b
a
g (t) dt .
(9 marks)
(b) (i)
Let n be a positive integer.
Using (a), or otherwise, show that there exists c   2n , (2n  1)

(2n  1)
2n

such that
cos x
 2n  1 
dx = (cos c)  ln
.
2n 
x

(ii) Hence, or otherwise, evaluate
lim
n 

(2n  1)
2n
cos x
dx .
x
(6 marks)
END OF PAPER