QUEEN’S COLLEGE
Mock Examination, 2008-2009
Pure Mathematics II
Date: 19-2-2009
Time: 1:30 pm - 4:30 pm
Secondary 7E,S
Instructions:
(1) Answer ALL questions in Section A and any FOUR questions from section B.
(2) All workings must be clearly shown.
(3) Unless otherwise specified, numerical answers must be exact.
(4) The diagrams in this paper are not necessarily drawn to scale.
FORMULA FOR REFERENCE
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B  sin A sin B
tan( A  B) 
tan A  tan B
1  tan A tan 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  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)
1
Section A:(40 marks)
Answer ALL questions in this section.
1.(a) Evaluate lim tan x tan 2 x .
x

4
(b) Prove that lim x sin
x 0
1
 0.
x
(6 marks)
2.
( x  a) 2

Let f ( x)   2 x 2  sin bx

x

(a) If
if x  0
if x  0
.
f is continuous at x = 0, show that b  a 2 .
(b) If furthermore, f (x) is differentiable at x = 0, find the values of a and b.
(6 marks)
3.
Let f : N  R be a function such that f(1) = 1 and for all m, n  N ,
f(n + m) = f(n) + f(m) +nm
(a) Show by induction that f(n) > 0 for all n  N .
Hence, or otherwise, show that f is injective.
(b) By expressing f(k + 1) – f(k) in terms of k, find f(n) in terms of n.
(7 marks)
4.(a) Find
2  x2
 1  x 3 dx .
2n
(b) Evaluate lim

8n 2  k 2
n k 1 8n 3
 k3
.
(7 marks)
2
5.(a) (i)
Find the constants a and b such that
4x 2
2
( x  1)
2

1
a
1
b



.
2
x  1 ( x  1)
x  1 ( x  1) 2
1
(ii) Using the substitution u  1
, evaluate
2x
(b) As shown in the figure, the vertical lines x 

1
6
1
1
16
1
dx .
2x
1
1
and x 
16
6
meets the curve
1

y  4 8 2   at the points A and B. Find the volume of the solid obtained by revolving
x

the shaded region about x-axis.
(7 marks)
y
A
B
O
6.
1
16
 1
Let  be a real number. Define y  1  
x

(a) Show that
1

y  4 8 2  
x

1
6
x

for x > 0.
dy
 y

dx x( x  1)
(b) For n = 1, 2, 3, …, show that
( x 2  x) y n1  2nx  (n   )y n  (n 2  n) y n1  0 ,
where y 0  y and y k  
dk y
dx k
for k  1.
(7 marks)
3
Section B:(60 marks)
Choose any Four questions.
Each question carries 15 marks
x 3  1000
where x  0 or x  10 .
3x
(a) For x  0 or x  10 , find f (x) .
7. Let f ( x) 

 3000 x 3  250
Show that f " ( x) 

.
x  1000
3 x x  1000
3x
(b) Determine the values of x for each of the following cases:
(i) f ( x)  0 ,
(ii) f ( x)  0 ,
(iii) f " ( x)  0 ,
(iv) f " ( x)  0 .
3

3

(2 marks)
3
(c) Find all relative extreme point(s) and point(s) of inflexion of
(3 marks)
f (x), if any. (2 marks)
(d) Find the asymptote(s) of the graph of f (x).
(5 marks)
(e) Sketch the graph
(3 marks)
f (x).

2
8.
Let I n   cos n tdt for n  0 .
0
(a) (i)
Evaluate I o and I1 .
n 1
I n2 for n  2 .
n
Hence, or otherwise, evaluate I 2n and I 2n1 for n  1.
(ii) Show that I n 
(8 marks)
(2 marks)
(b) Show that I 2n1  I 2n  I 2n1 for n  1.
1  2  4  6    ( 2n) 
(c) Let a n  be a sequence defined by a n 
2n  1 1  3  5    (2n  1) 
(i) Using (a) and (b), show that
2n 

  an 
for n = 1, 2, 3, …
2n  1 2
2
(ii) Evaluate lim
n
2
for n = 1, 2, 3, ….
 2  4  6    ( 2 n) 

.
2n  1 1  3  5    (2n  1) 
1
(5 marks)
4
ln x 
2
9.(a) Find lim   1 
.
x   x
x 
(2 marks)
(b) Let f ( x)  2  x  ln x where x > 0
(i)
Show that the equation f ( x)  0 has exactly two real roots.
(ii) Let a and b be the two real roots of the equation f ( x)  0 . If a < x < b , write down
the range of values of f (x) .
(6 marks)
(c) The sequence x n  is defined by xn1  2  ln xn and 0  a  x1  b , where a, b are the
roots of the equation x  2  ln x .
(i)
Prove that for all positive integers n, a  xn  b .
(ii) Find lim xn .
n
(7 marks)
10. Let a and b be positive numbers and a  b .
(a) Show that a abb  abba . When does the equality hold?
ab
(b) Using (a), or otherwise, show that 

 2 
(3marks)
a b
 abba . When does the equality hold?
(4 marks)
(c) Show that x x (1  x)1 x
1

for 0  x  1 . When does the equality hold?
2
ab
Hence deduce that a abb  

 2 
a b
where equality holds iff a = b.
(8 marks)
5
11. Consider the ellipse E : 4 x 2  y 2  4 .
 1  t 2 4t 
, t  R .
,
Let P be the point 
1 t 2 1 t 2 


(a) (i)
Prove that P is a point on the ellipse.
(ii) Find the slope of the tangent to E at P.
Hence show that the equation of the tangent to E at P is (1  t 2 ) x  ty  1  t 2 .
(iii) Find the coordinates of the foot of perpendicular, N, from origin to the tangent at P.
(9 marks)
 1  s 2 4s 
.
,
(b) Through N, a tangent to (E) is drawn touching the ellipse at Q
1 s2 1 s2 


2
t 1  2t
(i) Show that s  2
.
t 2


(ii) If OQ produced meets the ellipse (E) again at R, find the coordinates of R in terms of s.
(6 marks)
…End of paper…
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