S.7 Mock Exam. 10-11 / Pure Maths. 2 / p.1
2011-AL
Pure Mathematics
Paper 2
LIU PO SHAN MEMORIAL COLLEGE
(2010-2011)
MOCK EXAMINATION
SECONDARY SEVEN
PURE MATHEMATICS PAPER 2
Date : 22 - 2 – 2011
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 REFERENCE
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. 10-11 / Pure Maths. 2 / p.2
Section A (40 %)
Answer ALL questions in this section.
1
1.
 x x
(a) Evaluate lim 
 .
x0  sin x 
 (x  a) 2

(b) Let f (x)   2x 2  sin bx

x

(i)
if x  0
if x  0
.
If f (x) is continuous at x = 0, show that b  a 2 .
(ii) If furthermore, f (x) is differentiable at x = 0, find the values of a and b.
(7 marks)
2. Let
f : R+  R be a function satisfying the following conditions:
(1)
f (x y) = y f (x) + x f (y)
(2)
f ’(1) = 2 and
for all x, y R+.
f (1  h)
h0
h
lim
= 2.
  h 
(a) By expressing f (x + h) as f  x 1    , or otherwise, prove that f is differentiable for x > 0 and
  x 
that for all x  R+,
f ’(x) = 2 +
(b) Show that f ”(x) =
f (x)
.
x
2
.
x
Hence find f (x).
(6 marks)
3. Let
f : I+  R be a function such that f (1) = 1 and for all m, n  I+,
f (n + m) = f (n) + f (m) + nm.
(a) Show, by induction, that f (n) > 0 for all n  I+.
(b) Hence, or otherwise, show that f is injective.
(6 marks)
4. (a) Let x be a real number and m be a non-negative integer.
Define I (x) 
m
 2 cos  d
x
m
. Show that for m > 1,
1
m -1
cos m-1 x sin x 
I
( x) .
m
m
m m-2
 n
  k 
cos 6  
(b) Hence, or otherwise, evaluate lim
.

n 2n
2
2n


k =1
I
(x) 
(7 marks)
S.7 Mock Exam. 10-11 / Pure Maths. 2 / p.3
5. Given a hyperbola H:
(a) (i)
x 2 y2

 1 where a, b > 0.
a 2 b2
Prove that a line y = mx + c is a tangent to H if and only if b2 + c2 = a2m2.
(ii) Hence show that the line joining two points P(ap, -bp) and Q(aq, bq), where p, q are non-zero
constants, on the two asymptotes touches H if and only if pq = 1.
(b) Prove that the area of the triangle formed by the two asymptotes and a tangent of a hyperbola,
i.e. ∆OPQ in (a), is a constant.
(7 marks)
6. (a) (i)
Find the constants a and b such that
4x 2
1
a
1
b
.




2
2
2
x  1 (x  1)
x  1 (x  1) 2
(x  1)
(ii) Using the substitution u 

1
6
1
16
1
1
1
, show that
2x
1 3 7 
1
dx =  ln   .
4  2 12 
2x
(b) As shown in the figure, the vertical lines x 
1
1
1

and x 
meet the curve y  4 8  2  
16
6
x

at the points A and B. Find the volume of the solid obtained by revolving the shaded region
about the x-axis.
y
A
B
O
1
16
1
6
1

y  4 8 2  
x

x
(7 marks)
S.7 Mock Exam. 10-11 / Pure Maths. 2 / p.4
Section B (60 %)
Answer any FOUR questions in this section.
7. Let
f (x) =
3- 2 x
e
(a) (i)
1
x
where x  0.
Show that f (x) is an even function.
(ii) For x > 0, find f ’(x) and show that f ”(x) =
3 - 8x
4
x e
1
x
.
(3 marks)
(b) For x > 0, find 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).
(3 marks)
(d) Find the asymptote(s) of the graph of y = f (x).
(4 marks)
(e)
Sketch the graph of y = f (x).
(2 marks)
8. (a) Show that

S.7 Mock Exam. 10-11 / Pure Maths. 2 / p.5

sec 2 x
4
dx = ln 2.
0 1  tan x
(2 marks)
(b) Let f (x) be a function continuous on [0 , a].
(i)
Show that

a
0

f (x) dx 
a
0
f (a - x) dx .
(ii) Suppose furthermore that f (x) = f (a – x) for all x  [0 , a]. Show that

a
0
x f (x) dx 
a
2

a
0
f (x) dx .
(5 marks)
(c) (i)
Using (b) (i), show that


 1
1
4
dx   ln 2 .
8 4
0 1  tan x
(ii) Hence, or otherwise, evaluate


tan 2 x
4
dx .
0 1  tan x
(5 marks)
(d) Evaluate


2
4 x sec x dx .
0 1  tan x
(3 marks)
S.7 Mock Exam. 10-11 / Pure Maths. 2 / p.6
9. (a) Show that

1
ln (1  x) dx  2 ln 2 - 1 .
0
(3 marks)
(b) Let f (x) be an increasing function on [0 , 1] and n be a positive integer.
(i)
Show that
1  k 1
f
 
n  n 

k
n
f (x) dx 
k -1
n
1
f
n
k
 
n
for k = 1, 2, …, n.
(ii) Hence deduce that
n
1  k 1
 n f  n  
k =1

1
0
n
1 k
f  .
k =1 n  n 

f (x) dx 
(5 marks)
(c) (i)
Hence, show that
1


n


1
2n
!



ln 
 
n  2  n ! 


(ii) Deduce that

4
ln

e
1


n


1
2n
!



ln 
 .
n  n !  


1


n
4


1
2n
!
 
 
lim
= .

n  n  n !  
e


(7 marks)
 x  at 2
10. Consider the curve P: 
 y  2at
where t  R.
2
2
Let A  at 1 , 2at 1  and B  at 2 , 2at 2  be two distinct points on P, where t1and t2 are non-zero real




numbers.
(a) (i)
Find the equation of the normal to P at A.
(ii) Prove that AB is normal to P at A if and only if t 1  t 1 t 2  2  0 .
2
(6 marks)
(b) Let F = (a, 0). It is given that the chord of P joining A and F cuts the curve P again at the point C.
(i)
Find the y-coordinate of C in terms of a and t 1 .
(ii) Suppose D is the point of intersection of the tangents at A and B. AB is also normal to P at A.
(1) Show that CD is horizontal.
(2) Show that the mid-point of AD lies on the line x = -a.
(3) Find the equation of the locus of D as t 1 varies.
(9 marks)
S.7 Mock Exam. 10-11 / Pure Maths. 2 / p.7
11. (a) Let f (x) be a twice differentiable function defined on (0,  ) where f ”(x)  0 for x > 0.
(i)
Let a and b be two distinct positive real numbers. Let c lies on the open interval with end points
a and b. Using the Mean Value Theorem, show that
(c – a) f ’(c)  f (c) – f (a)
and
(c – b) f ’(c)  f (c) – f (b).
(ii) For any positive real numbers r, a and b, show that
 ar  b 
f
 
 1 r 
r f (a)  f (b)
.
1 r
(9 marks)
(b) Let x , x , ... , x n be distinct positive real numbers.
1 2
(i)
Using (a), show that for all positive integers n  2,
 x  x2  xn
ln  1

n

ln x  ln x    ln x

1
2
n .
 

n

(ii) Show that for all positive integers n  2,
x + x ++ x
1
2
n
n
 n x x x .
1 2
n
(6 marks)
END OF PAPER