2007-AL
P MATH
KIANGSU-CHEKIANG COLLEGE (SHATIN)
FINAL EXAMINATION 2006-2007
F.6 PURE MATHEMATICS
PAPER 2
Monday, June 25, 2007
11.05 am  1.20 am (2 hours 15 minutes)
This paper must be answered in English
1.
This paper consists of Section A and Section B.
2.
Answer ALL questions in Section A and any THREE questions in Section B.
3.
Write your answers in the answer book provided.
4.
Unless otherwise specified, all working must be clearly shown.
©
S.M Fan
All Rights Reserved 2007
2007-F.6-P MATH 2-1
FORMULAS 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)
2007-F.6-P MATH 2-2
Section A (35 marks)
Answer ALL questions in this section.
Write your answers in the answer book.
1.
(a)
Prove that lim x ln x  0 .
(b)
Let h, k be real constants and f : R  R be defined by
x 0
h sin x  k cos 2 x  1
f (x) =  2
 x ln x
when x  0 ,
when x  0 .
It is given that f (x) is differentiable at x = 0, find the values of h and k.
(6 marks)
2.
Let F(u) =
Show that
z
z
u
0
x
0
f ( t ) dt , where f (t) is a continuous function.
F( u 2 ) du 
z
x2
0
( x  u ) f ( u ) du for all x > 0.
(5 marks)
3.
| x  1|
Let f : R  R be defined by f (x) = 
0
when 0  x  2
otherwise
(a)
Sketch the graph of y = f (x).
(b)
Is f a surjective function? Explain your answer.
(c)
Let g : R  R be defined by g(x) =f (x  l) f (x + l) + 1.
(i)
Is g an even function?
(ii)
Sketch the graph of y = g(x) .
.
(7 marks)
2007-F.6-P MATH 2-3
4.
Suppose f (x) and g (x) are real-valued continuous functions on [0, a] satisfying the
conditions that f (x) = f (a  x) and g (x) + g (a  x) = K where K is a constant.
1
a
a
0 f ( x) g ( x)dx 2 K 0 f ( x)dx .
Show that
Hence , or otherwise, evaluate

0 xsin xcos
4
xdx .
(6 marks)
5.
Evaluate
1
 1  x 3 dx
.
(5 marks)
6.
(a)
Let
f (x) = xx
for all x > 0. Prove that
f '(x) = xx (1 + ln x).
xx  x
.
x 1 ln x  x  1
lim
Hence evaluate
x
(b)
Evaluate
lim
x 0
0 t sin(sin t )dt .
x3
(6 marks)
SECTION B (45 marks)
Answer any THREE questions in this section. Each question carries 15 marks.
Write your answers in the answer book.
7.
Let f (x) =
( x  6)( x  8)2
( x  15)2
(x  15) .
(a)
Find f '(x) and f "(x) .
(b)
Solve each of the following inequalities:
(i)
f '(x) > 0,
(3 marks)
(ii)
f "(x) > 0.
(2 marks)
(c)
(d)
(e)
Find the relative extreme point(s) and point(s) of inflexion of the graph of
y = f (x)
(4 marks)
Find the asymptote(s) of the graph of y = f (x).
(3 marks)
Sketch the graph of y = f (x) .
(3 marks)
2007-F.6-P MATH 2-4
8.
Let g : R+  R be a twice differentiable function such that g " (x) < 0  x > 0.
(a)
Let a and  be real constants such that  > 0. Show that the function
 x   a  g ( x)   g (a )
F ( x)  g 

1 
 1  
attains its least value when x = a.
(4 marks)
(b)
Let 1, 2, ...., m (m  2) be m positive real numbers.
(i)
Prove by mathematical induction , or otherwise, that
 1x1  2 x2  ...  m xm  m
k g ( xk )
g

1  2  ...  m

 k 1 1  2  ...  m
for all x1, x2 , ..., xm > 0
(ii)
By considering g (x) = ln x, or otherwise, deduce that
a1  a2  ...  am m
 a1a2 ...am
m
for any positive numbers a1, a2, ..., am.
(11 marks)
9.
For any non-negative integer n, let I n 
(a)
Prove that

4
0

tan n d .
(i)
I n  I n 1  0
(ii)
I n  I n 2 
(iii)
1
1
 In 
2(n  1)
2(n  1)
1
n 1
for n  0 ,
for n  2 ,
for n  2 .
(8 marks)
(b)
For n  1,2,, let
an 
n

k 1
( 1) k 1
.
k
Using (a)(ii), or otherwise, express I2n +1 in terms of an. Hence use (a)(iii) to
evaluate lim an .
n 
(7 marks)
2007-F.6-P MATH 2-5
10.
(a)
Suppose that 0  a < b. Using Mean Value Theorem, or otherwise, prove that
1
tan 1 b  tan 1 a
1
.


2
1 b
ba
1  a2
(b)
(4 marks)
3
 1 2 .
8
(i)
Prove that tan
(ii)
Using (b)(i), prove that tan

24
 6  2  3  2.
(7 marks)
(c)
Using (a) and (b)(ii), prove that 3( 6  2)    24( 6  2  3  2) .
(4 marks)
END OF PAPER
2007-F.6-P MATH 2-6