Download formula for reference

QUEEN’S COLLEGE
Half-yearly Examination, 2008-2009
Pure Mathematics
Date: 9-1-2009
Time:8:30 – 11:30 am
Secondary 6E,S
Instructions:
(1) Answer ALL questions in Section A and section B.
(2) All workings must be clearly shown.
(3) Unless otherwise specified, numerical answers must be exact.
(4) The diagrams in this paper are not necessarily drawn to scale.
FORMULA 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)
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Section A:
(80 marks)
Answer ALL questions in this section.
1.
Let

1 5
,
2

1 5
.
2
 n  n 
Using Mathematical Induction, prove that
2.
(a) Resolve
2x  1
2
x x  1
2


(b) Hence find
3.
k 1
,n
(5 marks)
2k  1
.
2
k k  1
(6 marks)
2
It is given that f(x) = 2x 4  x 3  8x 2  5x  2 , g(x) = x 3  2x 2  5x  6 .
Let d(x) be the H.C.F. of f(x) and g(x).
(a) Using Euclidean Algorithm, or otherwise, find d(x) .
(b) Find polynomials u(x) and v(x) of degree less than 2 such that
d(x) = u(x) f(x) + v(x)g(x) .
Let
{an } be a sequence of positive integers such that

r 1
Prove by induction that an = 2n – 1 ,  n 
5.
6.
1 a n 
ar  

 2 
.
(6 marks)
(5 marks)
2
for n 
.
(10 marks)
Show, by multiple root theorem, that the polynomial equation
f(x) = 8x4 + 84x3 + 114x2 + 55x + 9 = 0
has a root of multiplicity 3 .
(10 marks)
Show, by Binomial Theorem, that
(a) C1n  2Cn2  3C3n  ...   1 nC nn  0
n 1
(5 marks)
(b) C 0n C nr  C1n C nr 1  C n2 C nr  2  ...  C nn r C nn 
(c)
7.
(12 marks)
into partial fractions .
n
4.
.
C 0n C1n  C1n C n2  C n2 C3n  ...  C nn 1C nn 
(a) Find the constants

A, B
2n 2n  12n  2...n  r  1
.
n  r !
22n  1!
n  1 n  1!2
(4 marks)
such that
 

(5 marks)
in terms of n .
(6 marks)
x 5  A x  1 x 3  x  1 x 3  B x  1 x 2  x  1 x 2
3
(6 marks)
3
2
2
n
(b) Hence, express in factorized form:

k5
k 1
2
Section B:
(120 marks)
Each question carries 30 marks
8.
(a) Solve the inequality
(b) Let
x 8  x 2.
(8 marks)
f(x) = x 3  4x 2  4x  1 , find
Hence solve the inequality :
surds.
a rational root for
f(x) = 0.
x 3  4x 2  4x  1 > 0 , leave your answer in terms of
(12 marks)
(c) Use (a)
and
(b)
to solve the inequality :
x  4x  2x  2  6  x  4x  2x  2  12
(10 marks)
9.
Let
, ,  be the roots of the cubic equation
Suppose p, q, r  0 and
(a) Show that
x 3  px 2  qx  r  0 .
<<.
   2  p  q .
(7 marks)
(b) Use (a) to find a cubic equation whose roots are
   2 ,    2 ,    2 .
(7 marks)
(c) Hence, or otherwise, show that the necessary and sufficient condition such that the roots
, ,  form a geometric sequence is rp 3  q 3  0 .
(6 marks)
x 3  2x 2  4x  r  0
(d) If
, ,  ( <  < ) are roots of the cubic equation
and , ,  form a geometric sequence , find  , 2 + 2 + 2 and 3 + 3 + 3 .
(10 marks)
10. (a) By considering the function : f(x) = (a1x + b1)2 + (a2x + b2)2 + ..... + (anx + bn)2
 n a 2 
k1 k 
prove the Schwarz’s inequality :
where
(b) Let
 n b 2    n a b 
k1 k  k1 k k 
ak, bk (k = 1,2,..., n) are non-zero positive real numbers.
xk, yk
(k = 1,2,...,n) are non-zero positive real numbers, use
 n x k  n
 n 
    x k y k    x k 
 k 1 
k 1 y k k 1
2
(7 marks)
(a) to show that :
2
(7 marks)
3
(c) (i)
,  so that for all real numbers
Find
k:
k(k + 1) =  k(k + 1)( k+ 2) - (k – 1)k(k + 1)
(4 marks)
(ii) Hence show that for all
n = 1, 2, 3, ....
1
1  2  2  3  ...  n n  1  n n  1n  2
3
(4 marks)
(d) Deduce that for all natural number n :
1 2 3
n 1
n
3n n  1
  ... 


2 3 4
n
n  1 4n  2
11. Let

(8 marks)

n
1  x  x 2  a 0  a1x  a 2 x 2 ... a r x r ... a 2 n x 2 n
(a) Show that :
(i)
a 0  a 1  a 2  ...  a r  ...  a 2 n  3n
(ii) a 0  a1  a 2  ...   1 a r  ...  a 2n  1
r
(4 marks)
(b) Use (a) to evaluate :
(i)
a 0  a 2  ...  a 2 r  ...  a 2 n
(ii) a1  a 3  ...  a 2 r 1  ...  a 2 n 1
(c) Find, in terms of
(6 marks)
a1  2a 2  3a 3  ...  ra r  ...  2na 2n
n:
(d) Obtain an expansion by replacing x
Hence show that
(e) Using the identity
evaluate
ar
=
1  x
a2n-r ,
2
by
(6 marks)
1
.
x
r = 0, 1, ... , 2n.
(6 marks)
 x 4   1  x  x 2  1  x  x 2  ,
n
a 02  a12  a 2 2 ... 1
n
n 1
a n 12
in terms of
n
an .
(8 marks)

End of paper 
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