2005 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
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)
Section A
1. For each positive integer n, define Sn  (1  5 )n  (1  5 )n .
Prove that
(a) S n2  2S n1  4S n ,
(b)
S n is divisible by 2 n .
(6 marks)
2. For any two positive integers k and n, let Tr be the rth term in the expansion of (1  x) kn in ascending
powers of x, i.e. Tr  Crkn1 x r 1 .
(a)
Suppose x 
2
. Find, in terms of k and n, the range of values of r such that Tr 1  Tr .
k
(b)
Suppose x 
2
. Using the result of (a), find the greatest term in the expansion of (1+x)51.
3
(6 marks)
3. (a)
By considering the function f ( x)  x  ln( x  1) , or otherwise,
prove that x  ln( x  1) for all x  1 .

(b)
Using (a), prove that the series
1
n
is divergent.
n 1
(7 marks)
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4. Let f(x) be a polynomial of degree 4 with real coefficients. When f(x) is divided by x  2 , the
remainder is 4. When f(x) is divided by x  3 , the remainder is  6 . Let r(x) be the remainder when
f(x) is divided by ( x  2)( x  3) .
(a)
(b)
Find r(x).
Let g( x)  f ( x)  r ( x) . It is known that g(x) is divided by x 2  1 and g (1)  16 . Find g(x).
(7 marks)
 3

5. Let T1 be the transformation which transforms a vector x to a vector y = Ax, where A   2
1

2
0
(a) (i) Find y when x =   .
 2
1

2 .
3

2 
(ii) Describe the geometric meaning of the transformation T1.
(iii) Find A2005.
(b) For every integer n greater than 1, let Tn be the transformation which transforms a vector x to a
vector y = Anx.
 2 
0
?
Is there a positive integer m such that the transformation Tm transforms   to 


2
 
 2 
Explain your answer.
(7 marks)
6. Let θ be a real number.
(a)
(b)
7. (a)
Solve the quadratic equation z 2  2 z cos 6  1  0 .
Using the result of (a), express x 6  2 x 3 cos 6  1 as a product of quadratic polynomials with
real coefficients.
(7 marks)
Consider the system of linear equations in x, y, z
 x
ay 
z b

( E ) : 2 x  (a  3) y  (a  1) z  0 ,
3 x 
a 2 y  (4a  1) z  b

where a, b  R.
(i) Find the range of values of a for which (E) has a unique solution. 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 = 1,
(2) a =  2 .
(10 marks)
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z 
b
 x  2y 

(b) Suppose that a real solution of 2 x 
y  3z 
0 satisfies x 2  y 2  z 2  b  3 ,
3x  4 y  7 z   b

where b  R. Find the range of values of b.
(5 marks)
8. (a)
 p q
 , where p , q , r , s R.
Let M  
r s 
(i) Suppose det M  0 .
Prove that M n1  ( p  s) n M for any positive integer n.
(ii) Suppose qr  0 . Let  and  be the roots of the quadratic equation
x 2  ( p  s) x  det M  0 .
Denote the 2  2 identity matrix by I.
(1) Prove that  and  are two distinct real numbers.
 0 0
 .
(2) Prove that M 2  (   ) M   I  
 0 0
(3) Define A  M   I and B  M   I .
 0 0
 and det A  det B  0 .
Prove that AB  BA  
 0 0
Find real number  and  , in terms of  and  , such that M  A  B .
(11 marks)
n
(b)
1 2 
 , where n is a positive integer.
Evaluate 
 4 3
Candidates may use the fact, without proof, that
 0 0
 ,
If X and Y are 2  2 matrices satisfying XY  YX  
0
0


n
n
n
Then ( X  Y )  X  Y for any positive integer n.
(4 marks)
9. Let a1 and b1 be real numbers satisfying a1b1  0 .
For each n = 1, 2, 3, …, define
a  bn
an1  n
an  bn
2
(a)
2
and bn1 
2anbn
.
an  bn
Suppose a1  b1  0 .
(i) Prove that an  bn for all n = 1, 2, 3, … .
(ii) Prove that the sequence an  is monotonic decreasing and that the sequence bn  is
monotonic increasing.
(iii) Prove that lim an and lim bn both exist.
n
n
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(iv) Prove that lim an = lim bn .
n
n
(v) Find lim (an  bn ) and lim an in terms of a1 and b1 .
n
n
(12 marks)
(b)
Suppose a1  b1  0 .
Do the limits of the sequences an  and bn  exist? Explain your answer.
(3 marks)
10. (a)
Let f ( x)  x 4  2ax 2  4bx  c , where a, b, c  R with b  0 . It is known that
f ( x)  ( x 2  2tx  r )( x 2  2tx  s) , where r, s, t R.
(i) Prove that t  0 .
(ii) Express r and s in terms of a, b and t.
(iii)Prove that 4t 6  4at 4  (a 2  c)t 2  b 2  0 .
(6 marks)
(b)
Consider the equation y 4  4 y 3  2 y 2  52 y  9  0 ……(*).
(i) Find a constant h such that when y  x  h , (*) can be written as
x 4  8 x 2  64 x  48  0 ……(**).
(ii) Using the results of (a), solve (**) in (b)(i).
Hence write down all the roots of (*).
(9 marks)
11. (a)
For any positive integer n, prove that t n  1  n(t  1) for all t > 0.
(3 marks)
(b)
(i) Let a, b and c be positive real numbers.
abc 3
2ab
abc

 abc  
 ab  .
in (a), prove that
3
3 2
ab

be positive real numbers, where k is a positive integer. Using (a), prove
By putting n = 3 and t 
(ii) Let y1 y 2 , …, y k 1
3
that yk 1  (k  1)Gk 1  kGk , where Gk  k y1 y2 ... yk and Gk 1  k 1 y1 y2 ... yk 1 .
(iii)Using mathematical induction and (b)(ii), prove that
x1  x2    xn n
 x1 x2  xn for any n positive real numbers x1 , x2 , , xn .
n
(8 marks)
(c)
Let n be a positive integer. Using (b)(iii), prove that n n  (1)(3)(5)(2n  1) .
Hence prove that (n 2  n) n  (2n)!
(4 marks)
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12. (a)
Let a, b and c be vectors in R³.
a1
a2
a3
(i) Prove that a  (b  c) = b1
c1
b2
b3 , where a = a1 i + a 2 j + a3 k, b = b1 i + b2 j + b3 k and
c3
c2
c = c1 i + c2 j + c3 k. Hence deduce that a  (b  c) = b  (c  a) = c  (a  b) .
 x  (b  c)   x  (c  a)   x  (a  b) 
a  
b  
c for
(ii) Suppose a  (b  c)  0 . Prove that x  
 a  (b  c)   b  (c  a)   c  (a  b) 
any vector x in R³.
(iii) Suppose a  a  b  b  c  c  1 and a  b  b  c  c  a  0.
(1) Prove that a  (b  c)  1 .
(2) Using (a)(ii), prove that x  (x  a)a  (x  b)b  (x  c)c for any vector x in R³.
(12 marks)
(b)
1
1
(i  j  k ) , v 
(i  k ) , w 
1
(i  2 j  k ) and r  6i  j  10k .
3
2
6
Find real numbers  ,  and  such that r =  u +  v +  w.
(3 marks)
Let u 
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