Download S6 Test 1

S6 Pure Mathematics Test 1
1.
Given f(x) = 16x4 +24x3 + 24x2 + 6x – 1. Express f(x) as a polynomial in (2x + 1).
2.
Resolve
3.
Factorize f(a,b,c) = a3(b - c) + b3(c - a) + c3(a - b).
4.
(a) Resolve
x 5  3x 3  4
into partial fractions.
x 10 ( x 2  1)
1
x ( x  1)( x  2)
4 – 10 - 2000
(6 marks)
(8 marks)
(6 marks)
into partial fractions.
n
1
k 1 k ( k  1)( k  2)
(b) Evaluate lim 
5.
(a) Show that, if a polynomial F(x) is divisible by (x + a)n , where n is a positive integer,
then its derivative F’(x) is divisible by (x + a)n-1 .
(4 marks)
(b) Let g(x) be a polynomial of degree 3 such that g(x) + 1 is divisible by (x – 1)2 and g(x) – 1
is divisible by (x + 1)2 . Using the result in (a), or otherwise, find the polynomial g(x).
6.
(7 marks)
(5 marks)
(a) Let a1 ,a2, …….an be distinct real numbers. Suppose f(x) is a polynomial of degree less
(AL85)
f ( x)
than n – 1 and the expression
is resolved into partial fractions
( x  a1 )( x  a 2 ).........( x  a n )
cn
c1
c2
as
. Show that c1 + c2 +….+cn = 0
(6 marks)

 ........ 
( x  a1 ) ( x  a 2 )
(x  an )
px  q
(b) Let F ( x) 
be resolved into partial fraction as
( x  a)( x  a  1)( x  a  2)
b3
b1
b2


x  a x  a 1 x  a  2
Show that for N > 3,
N
b  b3
b3
b
b  b2
F (k )  1  1
 2

(8 marks)

1 a 2  a
N  a 1 N  a  2
k 1