Download 2003 Pure Maths Paper 1

2003 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) Solve the inequality
x  6  3 , where x is a real number.
(b) Using the result of (a), or otherwise, solve the inequality 1  2 y  6  3 , where y
is a real number.
(6 marks)
2.
For any positive integer n, let C kn be the coefficient of x k in the expansion of
1  x n . Evaluate
(a)
C 1n  C 2n  C 3n    C nn ,
(b) C 1n  2C 2n  3C 3n    nC nn ,
(c)
C 1n  22 C 2n  32 C 3n    n 2C nn .
(6 marks)
3.
5x  3
into partial fractions.
xx  1x  3
n
5k  3
3

(b) (i) Prove that 
for any positive integer n.
2
k 1 k k  1k  3
(a) Resolve

(ii) Evaluate
5k  3
 k k  1k  3 .
k 1
(7 marks)
4.
Let
xn 
be a sequence of positive real numbers, where x1  2 and
xn 1  xn  xn  1 for all n = 1, 2, 3, …
2
n
1
for all n = 1, 2, 3, …
i  1 xi
(a) Using mathematical induction, prove that for any positive integer n,
(i) xn  n ,
1
(ii) S n  1 
.
xn 1  1
Define S n  
(b) Using (a), or otherwise, prove that lim S n exists.
n 
(7 marks)
5.
u   n  (1   ) m
Let m and n be vectors in R³ and  R. It is given that 
v  2(1   ) n   m
(a) Prove that u  v  32  4  2 m  n.


(b) Suppose∣m∣= 4 ,∣n∣= 3 and the angle between m and n is

.
6
(i) Evaluate∣m  n∣.
(ii) Find the smallest area of the parallelogram with adjacent sides u and v
as  varies.
(6 marks)
6.
(a) Suppose the cubic equation x 3  px 2  qx  r  0 , where p, q and r are
real numbers, has three real roots.
Using relations between coefficients and roots, or otherwise, prove that the
three roots form an arithmetic sequence if and only if
p
is a root of the
3
equation.
(b) Find the two values of p such that the equation x 3  px 2  21x  p  0 has three
real roots that form an arithmetic sequence.
(8 marks)
Section B
7. (a) Consider the system of linear equations in x, y, z
 x  ay 
z 0

(E) :  2x 
y
az   2a , where a R.
 x  2a 2 y  (a  3) z  2a

(i) Find the range of values of a such that (E) has a unique solution. Solve
(E) when (E) has a unique solution.
(ii) Solve (E) for
(1) a  1 ,
(2) a  4 .
(10 marks)
 x y z 0

(b) Suppose (x, y, z) satisfies  2 x  y  az   2 .
 x  2 y  2 z  2

Find the least value of 24 x 2  3 y 2  2 z and the corresponding values of x, y, z.
(5 marks)
8.
(a)
 2 
3
 =0, find the two values of  .
If det 
 3





(2 marks)
(b) Let  1 and  2 be the values obtained in (a), where  1 <  2 .
  2  1
3

Find 1 and  2 such that 
 3
  1 

 2 2
3


 3



2

 cos 1
Let P  
 sin 1
 cos 1   0 

    , 0  1   ,
 sin 1   0 
 c o s 2   0 

    , 0   2   .
 s i n 2   0 
cos  2 
 . Evaluate P n , where n is a positive integer.
sin  2 
 2
Prove that P 1 
 3
3
 P is a matrix of the form
0 
 d1

0
0
.
d 2 
(8 marks)
 2
(c) Evaluate 
 3
n
3
 , where n is a positive integer.
0 
(5 marks)
9.
(a) Consider the vectors a = (p, q, 0), b = (q, -p, 0) and c = (0, 0, r), where p, q and r
are non-zero real numbers.
(i) Prove that a, b and c are linearly independent.
 d a   d b 
 d c 
(ii) Let d be a vector in R³. Prove that d   2  a   2  b   2  c .
 a   b 
 c 

 



(6 marks)
(b) Let x, y and z be linearly independent vectors in R³.
 y u 
Define u = x and v  y   2  u .
 u 


(i) Prove that v is a non-zero vector.
 z u 
zv
(ii) Define w  z   2  u   2  v .
 u 
 v 




(1) Prove that u, v and w are orthogonal.
(2) Describe the geometric relationship between w and the plane containing
the vectors x and y.
(9 marks)
10. (a)
Let a and b be non-negative real numbers. Prove that a  b   a n  na n 1b for
n
all n = 2, 3, 4, … Write down a necessary and sufficient condition for the equality
to hold.
(3 marks)
(b) Let a1 , a2 , a3 ,  be a sequence of positive real numbers satisfying
a1  a2  a3   .
For any positive integer n, define An 
a1  a2  a3    an
and
n
Gn  a1 a 2 a3  a n  n .
1
(i) Prove that Ak 1  Ak for all k = 1, 2, 3, …
k 1
(ii) Using (a), prove that Ak 1  Ak ak 1 for all k = 1, 2, 3, … . Hence prove
that An  Gn and An  Gn if and only if a1  a2  a3    an for all
n = 1, 2, 3, …
(8 marks)
k
n
(c)
n  2  n  1  n 1
Let n be a positive integer. Using (b), prove that

 .
n 1  n 
1 

Hence deduce that 1 

 n 1
n 1
n
 1
 1   .
 n
(4 marks)
11. (a) Consider the equation x 4  ax 2  bx  c ………… (*), where a, b and c are real
numbers.
(i) Suppose b  0 . Solve (*).
(ii) Suppose b  0 .




(1) Prove that (*) can be written as x 2  t  a  2t x 2  bx  c  t 2 ,
where t is any real number.
(2) Prove that there exists a real number t 0 such that the equation
a  2t0 x 2  bx  c  t0 2   0
2
has a repeated root.
Hence, deduce that (*) can be written as x 2  t 0   a  2t 0 x    for
some real number  .
(9 marks)
4
2
Consider the equation x  6 x  12 x  8 ………… (**).
Find a real value of t such that the equation 6  2t x 2  12 x  8  t 2  0 has a
repeated root.
Hence solve (**).
(6 marks)
2
(b)
2


12. Let n be a positive integer.
(a) (i) Find all the roots of z 2 n  1  0 .
(ii) By factorizing z 2n  1 into a product of quadratic factors with real
2k  1 
1 n 1 
1
coefficients, or otherwise, prove that z n  n    z   2 cos

z
2n 
z
k 0 
for all z  0 .
(7 marks)
(b) Using (a), or otherwise, prove that
n 1
2k  1   cos n for any

(i)   cos  cos
 R,

2n 
2 n 1
k 0 
(ii)
n 1
2k  1
k 0
4n
 cos

2
2n
.
(8 marks)