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Pui Ying College (99-00)
First Examination
F.6 Pure Mathematics
Time allowed : 3 hours
Name :
Class : 6 B
Note :
No.
There are 14 questions. Answer ALL questions.
1.
Factorize a 3  b3  c3  3abc.
Hence prove that if a + b + c  0, then
a 3  b 3  c3
 abc .
3
2.
(4 marks)
Given ,  and  are the roots of the equation x3  2x 2  1  0.
Find the values of:
(a)  +  + ,
(b) 2 + 2 + 2,
(c) 3 + 3 + 3.
3.
(5 marks)
Prove by contradiction that
2  3 is irrational.
(Hint : You may use the fact that
4.
Let
f ( x )  ln
2 is irrational.)
1
for x  1.
1 x
Prove , by Mathematical Induction, that f ( n ) ( x ) 
5.
Let
xn n0
(5 marks)
(n  1)!
(1  x )n
for n  1.
(4 marks)
be a sequence of real numbers satisfying
x 0  0, x1  7 and x n  2  12x n  x n 1 for n  0, 1, 2,...
Show by mathematical induction that for n  0, x n  (4) n  3n .
6.
If
(6 marks)
y = ex tan x, prove that
y  2(1  tan x ) y  (1  2 tan x ) y  0.
Hence find y (
3
).
4
(5 marks)
7.
If x = a cos t and y = b sin t, where t is a parameter, find
8.
Define a function  :   R by
Γ (n)  

0
d2y
dx 2
in terms of t.
(4 marks)
x n 1e x dx.
Prove that
(a) (n  1)  n(n ) and
(b) (n  1)  n!.
(5 marks)
F.6PM/1stExam99-00/LCK/P1. of 2
9.
Differentiate each of the following functions with respect to x:
(a)
y  x 2 2x
(b)
y  x sin x
(c)
y
( x  1)( x  2)
( x  3)( x  4)
(9 marks)
10. Evaluate the following limits:
(a)
 2 3 
lim 1   2 
x 
x x 
11. Resolve
x
(b)
sin 1 x
lim
x 0
x
(c)
lim x x
x 0
(9 marks)
x2
in partial fractions.
( x 2  x  1)( x  1) 2
Hence evaluate
x2
dx.
( x 2  x  1)( x  1) 2

(8 marks)
12. Let f : R  R be a continuous function.
(a) Show that

a
0
f (t  b)dt  
a b
0
b
f (t )dt   f (t )dt for all a, b  R .
0
(b) If f(x + y) = f(x) + f(y) for all x, y  R,

show that
x
0
x
f (t  1)dt  f (1)x   f (t )dt for all x, y  R.
0
Using (a), or otherwise, show that f(x) = f(1)x for all x, y  R.
(8 marks)
13. For non-negative integers k and m, define
1
F(k, m)   u k (1  u 2 )m du.
0
(a) Show that
1
,
k 1
F(k ,0) 
(i)
(ii) F(k, m) 
2m
F(k  2, m  1) for m  1.
k 1
2 m (m!)
(b) Show that F(k, m) 
.
(k  1)( k  3)...( k  2m  1)
(c) Using (b), prove that

π
2
0
cos 2m 1θ dθ 
[2m (m!)]2
.
(2m  1)!
(4 marks)
(4 marks)
(4 marks)
14. Evaluate :
(a)
 ln( 1  x
(c)

2
)dx
dx
3  2x  x
2
(b)
e
(d)

7
2
x
sin 2
x
dx
2
x 1
dx (Hint : Put x  2  t 2 , where t  0) (16 marks)
x2
END OF PAPER
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