2007-AL
P MATH
KIANGSU-CHEKIANG COLLEGE (SHATIN)
FINAL EXAMINATION 2006-2007
F.6 PURE MATHEMATICS
PAPER 1
Monday, June 25, 2007
8.30 am  10.45 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 1-1
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 1-2
Section A (35 marks)
Answer ALL questions in this section.
Write your answers in the answer book.
1.
Let m and n be two positive integers with m > n.
n 1
(a)
 Cn2nr1  22n
Show that
.
r 1
(b)
1
By considering the expression (1  x) m (1  ) n or otherwise, find
x
mk
 Ckmr Crn
where
m  n  k  m.
r 0
(5 marks)
2.
(a)
(i)
(ii)
Resolve
1
into partial fractions.
x( x  1)( x  1)
3x 2  1
Using differentiation, or otherwise, resolve 2
into
x ( x  1)2 ( x  1)2
partial fractions.
3k 2  1
.

2
2
2
k  2 k (k  1) (k  1)

(b)
Evaluate
(6 marks)
3.
(a)
(b)
Let R be the matrix representing the rotation in the Cartesian plane
anticlockwise about the origin by 60o.
(i)
Write down R and R6.
(ii)
Let
2 1 
 . Verify that A1RA is a matrix in which all
A = 
0
3


elements are integers.
Using the results of (a), or otherwise, find a 2  2 matrix M, in which all the
1 0
 .
elements are integers, such that M3 = I but M  I, where I = 
0 1
(7 marks)
2007-F.6-P MATH 1-3
4.
(a)
(b)
Let A be a 3  3 non-singular matrix. Show that
x3
1

det (A  xI) = 
det (A  x 1I).
det A
F0 1
0 0
Let A = G
G
H4 17
I
1J.
J
8K
0
(i)
Show that 4 is a root of det(A  xI) = 0 and hence find the other roots
in surd form.
(ii)
Solve det(A1  xI) = 0
(6 marks)
5.
Let f : R\{1}  R\{1} be a function defined by f ( x) 
(a)
Prove that f is a surjective function.
(b)
If furthermore, f is an injective function, find
(i)
(ii)
x 1
.
x 1
f 1(x);
  
f 1 f 1 f 1 ... f 1 ( x)
 .
n f 1 ' s
(6 marks)
6.
Solve the inequality | x  1 |  | x + 2 | > 2.
(5 marks)
2007-F.6-P MATH 1-4
SECTION B (45 marks)
Answer any THREE questions in this section. Each question carries 15 marks.
Write your answers in the AL(A) answer book.
7.
8.
Solve the following system of equations:
(b)
Find all possible values of p and q such that the following system of equations
is solvable :
 x yz 3

(II) :  x  2 y  z  1 .
(7 marks)
 x  y  pz  q

(c)
Find the solutions, if possible, of the system of equations
 x yz 3
 x  2 y  z  1

(III) : 
.
 x  y  pz  11
 x 2  y 2  z 2  21

(a)
(I) :
 x yz 3
.

 x  2 y  z  1
(2 marks)
(a)
(6 marks)
Prove that for any positive integer n,
(i)
1
1
1
1
n 1
;


 ... 

1 2 2  3 3  4
(n  1) n
n
(ii)
n
1
1  1  2   k  1 
(1  )n  2   1  1   ... 1 
;
n
n  n  
n 
k 2 k ! 
(iii)
 n 1 
Hence, or otherwise, deduce that 2  
  3 for n  3.
 n 
n
(8 marks)
(b)
Using (a) and Mathematical induction, prove that for any positive integer
n
n  6,
n
n
   n!   
3
2
n
(7 marks)
2007-F.6-P MATH 1-5
9.
Let a1 and a2 be real numbers and p, q be positive constants such that
For each n = 1, 2, 3, ... , define an2 
(a)
Prove that an+4  an+2 =
(b)
Suppose that a2  a1 .
(i)
(ii)
1 1
  1.
p q
an1 an
 .
p
q
1
(an+2  an).
q2
(3 marks)
Prove that
(1)
a2n+1  a2n1 ,
(2)
a2n+2  a2n ,
(3)
a2n  a2n1 .
Prove that lim a2 n 1 and lim a2 n both exist.
n 
n 
(9 marks)
(c)
Suppose that a2 < a1. Do lim a2 n 1 and lim a2 n exist? Explain your answer.
n 
n 
(3 marks)
Note: You are not supposed to find an explicitly in doing this question.
10.
Let A =
(a)
3 1I
F
G
H2 0 J
Kand let v denote a 21 matrix.
Find the two real values 1 and 2 of  with 1 < 2 such that the matrix
equation
(1)
Av = v
has non-zero solutions.
(b)
(c)
(4 marks)
Let v1 and v2 be non-zero solutions of (1) corresponding to 1 and 2,
v 
v 
respectively. Show that if v1 =  11  and v2 =  12  , then the matrix
 v21 
 v22 
v 
v
V =  11 12  is non-singular.
 v21 v22 
(4 marks)
1 0
(i)
Using (a) and (b), show that AV = V
.
0 2
F
IJ
G
H K
F3 1I
Hence or otherwise, evaluate G J, n  N.
H2 0 K
n
(ii)
(7 marks)
THE END
2007-F.6-P MATH 1-6