Download FORMULAS FOR REFERENCE

2008 Pure Maths Paper 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  tan B
1  tan A tan B
A B
A B
sin A  sin B  2 sin
cos
2
2
A B
A B
sin A  sin B  2 cos
sin
2
2
A B
A B
cos A  cos B  2 cos
cos
2
2
A B
A B
sin
cos A  cos B  2 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)
tan( A  B ) 
SECTION A (40 marks)
1.
Denote the coefficient of x k in the expansion of (1  x) n by C kn .
(a) Let m be a non-negative integer with m  n . Using
(1  x) m (1  x) n  (1  x) mn , prove that
m
C
k 0
m
k
C rn k  C rm  n for all
r  m , m  1 , , n .
99
(b) Using (a), or otherwise, evaluate
C
99
k
C k100
1
C
99
k
101
k
k 0
99
k 0
.
C
(6 marks)
2.
(a) Resolve
7x  9
into partial fractions.
x( x  1)( x  3)
(b) Express
 k (k  1)(k  3)
n
7k  9
in the form A 
k 1
B
C
D


, where
n 1 n  2 n  3
A , B , C and D are constants.

(c) Evaluate
7k  9
 k (k  1)(k  3) .
k 5
(7 marks)
3.
(a) Let a , b and c be three numbers such that a  b  c  0 . Prove that
a 3  b 3  c 3  3abc .
(b) Using (a), solve the equation
( x  2 2  5 3 ) 3  (3x  6 2  7 3 ) 3  8( x  4 2  6 3 ) 3  0 .
(6 marks)
4. Let A be the matrix representing the rotation in the Cartesian plane
anticlockwise about the origin by 120 .
(a) Write down the matrix A .
(b) For every positive integer n , let Tn be the transformation represented
by A n . It is given that T1 transforms the point P to the point ( 4 , 2 3 ) .
(i)
(ii)
(iii)
Find the coordinates of P .
Describe the geometric meaning of the transformation Tn .
Is there a transformation Tn which transforms P to the point
( 3 , 5 3 )? Explain your answer.
(7 marks)
5. Let S  z C: 3 z  2i  z  8  2i .
(a)
(b)
Prove that S is represented by a circle on the Argand diagram. Also find
the centre and the radius of the circle.
On the Argand diagram, P is the point representing the complex number
 7  17i and Q is a point representing any z  S . Find the longest
distance between P and Q.
(7 marks)
6. Let  ,  ,  and  be positive real numbers.
(a)
(b)
Using Cauchy-Schwarz’s inequality, or otherwise, prove that
(i) (       ) 2  4( 2   2   2   2 ) ,
(ii) ( 2   2   2   2 ) 2  (       )( 3   3   3   3 ) .
Using (a), prove that
    
4

3  3  3  3
.
2   2  2  2
Write down a necessary and sufficient condition for the equality to hold.
(7 marks)
SECTION B (60 marks)
7. (a) Consider the system of linear equations in x , y , z
 x  (a  2) y  (a  1) z  1

(E):  x 
3y 
z  b , where a , b  R .
3x 
2 y  (a  1) z  1

Prove that (E) has a unique solution if and only if a 2  4 . Solve (E)
when (E) has a unique solution.
(ii) For each of the following cases, find the value(s) of b for which (E)
is consistent, and solve (E) for such value(s) of b .
(1)
a2 ,
(2)
a  2 .
(11 marks)
2
2
2
(b) Find the greatest value of 2 x  15 y  10 z , where x , y and z are
(i)
real numbers satisfying
 x  4 y  3z  1

x  3 y  z  0 .
3x  2 y  z  1

(4 marks)
8.
Denote the 2  2 identify matrix by I .
 p q
 , where p , q and r are real numbers.
Let M  
r

p


(i)
(ii)
Evaluate M 2 .
Let s be a real number. Prove that if p 2  qr  1 , then for any
positive integer n ,
( M  sI ) 2 n 
( s  1) 2 n  ( s  1) 2 n
( s  1) 2 n  ( s  1) 2 n
M
I.
2
2
(7 marks)
 6 3
 .
(b) Let A  

1
2


(i)
(ii)
(iii)
  
   I , where  ,  ,  and 
Express A in the form 





are real numbers.
Evaluate A800 .
Using the result of (b)(ii), or otherwise, evaluate ( A 1 ) 800 .
(8 marks)
9.
Let
x n 
and
y n 
xn y n  xn y n
2
xn1 
xn  y n
2
be sequences of real numbers, where x1  2 , y1  8 and
2
2
xn  y n
xn  y n
2
and y n 1 
Prove that xn1  y n1 
for all n  1 , 2 , 3 ,
 ( xn  y n )( xn  y n )
3
(a)
2
3
( xn  y n )( xn  y n )
2
2
.
(2 marks)
(b)
(c)
Prove that
(i)
xn  y n ,
(ii) xn1  xn ,
(iii) y n 1  y n .
Prove that
x n 
and
y n 
(5 marks)
converge to the same limit.
(4 marks)
(d)
Prove that x n y n is independent of n .
Hence, or otherwise, find lim x n .
n 
(4 marks)
10. Let p(x) be a polynomial of degree k with real coefficients satisfying the
following conditions:
(1) p( x)  p( x  1)  x100 for all x R ;
(2) p(1)  1 .
(a)
Find k and the coefficient of x k in p(x).
(3 marks)
(b) Find p(0) and p (1) .
(2 marks)
(c) Prove that p( x)  p( x  1)  p( x  1)  p( x) for all x R.
Hence, prove that p(n)  p(n  1)  0 for all n N.
(4 marks)
(d) Prove that p( x)  p( x  1)  0 for all x R.
(e) Prove that
p(x) is divisible by xx  12x  1 .
(3 marks)
(3 marks)
11. (a) (i) Prove that 10 is an irrational number.
(ii) Using (a)(i), prove that if m and n are integers such that m  n 10  0 ,
then m = n = 0.
(4 marks)
(b) Prove that for any positive integer n, there exists a unique pair of positive

integers a n and bn such that 3  10
3 
(c)
10

n

n
 an  bn 10 and
 an  bn 10 .


n
(6 marks)
For every positive integer n, define xn  3  10 . Prove that for any
positive integer n, the greatest integer less than x2 n1 is an even number
and the greatest integer less than x 2 n is an odd number.
(5 marks)