Download 2004 Pure Maths Paper 1

2004 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  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
A B
A B
sin
2
2
2 sin A cos B  sin( A  B)  sin( A  B)
cos A  cos B  2 sin
2 cos A cos B  cos( A  B)  cos( A  B)
2 sin A sin B  cos( A  B)  cos( A  B)
Section A
1.
(a) Resolve
1
into partial fractions.
2 x  12 x  12 x  3
n
(b) Prove that
1
1
1
1
 2k  12k  12k  3  12  82n  1  82n  3
for all positive
k 1
integer n.

Hence or otherwise, evaluate
1
 2k  12k  12k  3 .
k 1
(6 marks)
2.
Let a n  be a sequence of positive real numbers, where a1  1 and a n 
12a n  1  12
a n  1  13
,
n = 2, 3, 4, …
(a) Prove that a n  3 for all positive integers n.
(b) Prove that a n  is convergent and find its limit.
(6 marks)
3.
(a)
Let R be the matrix representing the rotation in the Cartesian plane anticlockwise
about the origin by 60 .
(i)
(ii)
Write down R and R 6 .
2 1 
 . Verify that A 1 RA is a matrix in which all the elements
Let A  
0
3


are integers.
(b) Using the results of (a), or otherwise, find a 2  2 matrix M, in which all the
1 0
 .
elements are integers, such that M 3  I but M  I , where I  
0 1
(7 marks)
4.
Let f ( x)  x 3  px 2  qx  r , where p, q and r are non-zero real numbers.
(a) If f(x) is divisible by x 2  q , find r in terms of p and q.
(b) Suppose that f(x) is divisible by both x  a and x  a , where a is a non-zero real
number.
(i) Factorize f(x) as a product of three linear polynomials with real coefficients.
(ii) If f(x) and f(x + a) have a non-constant common factor, find p in terms of a.
(7 marks)
5.
Let n be a positive integer.
(a) Let a  0 .
(i)
If k is a positive integer, prove that a  a k  1  a k  1 .
(ii)
Prove that 1  a   2 n 1 1  a n  .
n
(b) Let x and y be positive real numbers. Using (a)(ii), or otherwise, prove that
xn  yn
x y
.

 
2
 2 
n
(7 marks)
6.
Let   R. For each n N, define x n  sin n   cos n  .
(a) Find a function f ( ) , which is independent of n, such that x n  1 x1  x n  2  f ( ) x n .
Also express f ( ) in terms of x1 .
(b) Suppose that x1 is a rational number. Using mathematical induction, prove that
x n is a rational number for every n.
(7 marks)
7.
(a) Consider the system of linear equations
 x  (a  2) y  az  1

(E) :  x 
2 y  4 z  1 , where a, b  R.
ax 
y  3z  b

(i)
(ii)
Prove that (E) has a unique solution if and only if a  2 and a  4 . Solve
(E) in this case.
For each of the following cases, determine the value(s) of b for which (E) is
consistent, and solve (E) for such value(s) of b.
(1) a = 2,
(2)
a = 4.
(10 marks)
 2z  1
 x

(b) If all solutions (x, y, z) of  x  2 y  4 z  1 satisfy k x 2  3  yz ,
2 x 
y  3z  2

find the range of values of k.
(5 marks)
8.
  k     k 
 , where  ,  , k  R with    .
Let A  
  k 
 k
1
Define X 
 A  I  and Y  1  A  I  , where I is the 2  2 identity
 
 
matrix.
(a) Evaluate XY, YX, X  Y , X 2 and Y 2 .
(4 marks)
(b) Prove that A   X   Y for all positive integers n.
n
n
n
(4 marks)
 5 4

(c) Evaluate 
 2 3
2004
.
(d) If  and  are non-zero real numbers, guess an expression for A
 ,  , X and Y, and verify it.
1
(4 marks)
in terms of
(3 marks)
9.
(a)
Let f(x) be a polynomial with real coefficients. Prove that a real number r is a
repeated root of f(x) = 0 if and only if f (r )  f (r )  0 .
(5 marks)
(b) Let g( x)  x  ax  bx  c , where a, b and c are real numbers. If a  3b , prove
3
2
2
that all the roots of g(x) = 0 are distinct.
(6 marks)
(c) Let k be a real constant. If the equation 12 x  8 x  x  k  0 has a positive
repeated root, find all the roots of the equation.
(4 marks)
3
2
10. Let n be a positive integer.
(a) Suppose 0 < p < 1. f
(i) By considering the function f ( x)  x p  px on (0,  ) , or otherwise, prove
that x p  px  1  p for all x > 0.
(ii) Using (a)(i), or otherwise, prove that a p b1 p  pa  (1  p)b for all a, b > 0.
(iii) Let a1 , a2 , a3 ,, an and b1 , b2 , b3 ,, bn be positive real numbers. Using
p
n
(a)(ii), or otherwise, prove that
a
i 1
p
i
bi
1 p
 n
  n

   ai    bi 
 i 1   i 1 
(b) Suppose 0 < s < 2. Let x1 , x2 , x3 ,, xn and y1 , y2 , y3 ,, yn
1 p
.
(9 marks)
be positive real
numbers. Prove that
1
1
(i)
 n s 2 s  2  n 2 s s  2
xi yi    xi yi    xi yi  ,

i 1
 i 1
  i 1

(ii)
 n s 2 s  n 2 s s   n 2  n 2 
  xi yi   xi yi     xi   yi  .


 


 i 1
 i 1
  i 1  i 1

n
(6 marks)
11. (a)
Let u  2i  j  k and v  3i  4 j  k be vectors in R 3 .
(i)
(ii)
Find all vectors w in R 3 such that w  u  9 and w  v  1 .
Find the vector w 0 which has the least magnitude among the vectors
obtained in (a)(i).
Prove that u, v and w 0 are linearly dependent.
(7 marks)
3
(b) Suppose that a and b are linearly independent vectors in R . Let c be a linear
combination of a and b, p = c a and q = c  b .
(i)
For any vector r in R 3 satisfying r  a = p and r  b = q, prove that r  c
is orthogonal to both a and b.
Hence deduce that r  c .
(ii)
If c = sa + tb, express s and t in terms of p, q, a and b.
(8 marks)
12. Let n be a positive integer.
(a) Assume that  is not an integral multiple of  .
n
2  cos n  1  i sin n  1  .

2
2 
sin 
2
cos   i sin  n  1  sin
(i)
Prove that
(ii)
Using the identity
cos   i sin   1
 z n 1
k

 for z  1 , or otherwise, prove that
z

z

k 1
 z 1 
n
n
 cos k 
sin
k 1
n  1
n
cos
2
2
and

sin
2
n
 sin k 
k 1
n
(iii)
Using (a)(ii), or otherwise, prove that
sin
 cos
k 1
2
(k ) 
n  1
n
sin
2
2
.

sin
2
n sin n cos( n  1)

.
2
2 sin 
(9 marks)
(b) For n > 1, evaluate
n
 k 
(i)  sin 2 
,
 n 
k 1
k 
 k
 cos  .
 sin

n
n 
k 1 
n
(ii)
2
(6 marks)