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SAMPLE QUESTIONS:
1

ii)
1 3

1.
Rationalise:
2.
Solve for x :
3.
Solve for x :
4.
Solve the symmetric equation using a suitable substitution. x 4  5 x 3  10 x 2  10 x  4  0
5.
Solve the equations:
1 2  3
1  x 2
2  x2

3 1
3
1  a 2
2  a2
x 2  6 x  2 x 2  6x  24
i)
x  2y
y  2z
2x  z


, x yz2
3
4
5
ii)
x2
1
1
x

 1,
 3
4 y 1
3( y  1) 2
iii)
x
y

1
ya xa
x  y  2a
6.
i)
ii)
iii)
Solve the simultaneous equations using row reduction to echelon form.
7.
Solve for x :
8.
Find the value of k so that the equation 4 x 2  8 x  k  0 has equal roots.
9.
Find the values of  for which 10 x 2  4 x  1  2x2  x  has equal roots.
10.
Given that one root of the equation x 2  px  q  0 is twice the other, show that
x  3 y  3z  4,3x  y  2 z  1,2 x  y  z  7
2 x  5 y  2 z  14,9 x  3 y  4 z  13,7 x  3 y  3z  3
x  4 y  2 z  0,2 x  y  z  0,8 x  5 y  6 z  6
 
 
33 x  6 32 x  11 3 x  6  0
2 p 2  9q , hence, find the values of k , if the equation


x 2  2k  2x  k 2  3k  2  0 , has one root twice the other.
11.
If  and  are the roots of ax 2  bx  c  0 ax 2  bx  c  0 , obtain an equation
whose roots are
12.
1

3
and
1

3
. If in the above equation  2  1 , prove that a 3  c 3  abc  0 .
If  and  are the roots of ax 2  bx  c  0 , and  :  =  :  , show that
b 2  (   ) 2 ac.
13.



If  and  are the roots of x 2  px  q  0 , express    2    2 in terms of p
and q . Hence, deduce that for one root to be square the other, then;
p 3  3 pq  q 2  q  0 .
14.
When a polynomial f (x ) is divided by ( x  2) the reminder is -2, and when it is divided by
( x  3) the remainder is 3. Find the remainder when f (x) is divided by ( x  2)( x  3) .
2
A polynomial function Px has a factor of x  3x  4 and leaves a remainder of 5
when divided by x  2 . Determine the remainder when the polynomial is divided by
15.
x
2
16.

 3x  4  x  2 .
The polynomial 5 x 3  px 2  qx  r has a factor ( x  2) and a remainder of (3 x  1) when
divided by ( x 2  1) . Find the values of p, q and r .
17.
If 4 x 3  kx2  px  2 is divisible by x 2  a 2 , prove that kp  8  0 .
18.
Given that the polynomial f ( x)  Q( x) g ( x)  R( x) where Q (x ) is the quotient,
g ( x)  ( x   )( x   ) and R (x ) is the remainder, show that
R( x) 
( x   ) f ( )  (  x) f (  )
, when f (x ) is divided by g (x ) .
(   )
Hence, find the remainder when f (x ) is divided by x 2  9 , given that f (x ) divided by
( x  3) is 2 and when divided by ( x  3) is -3.
Given that x 4  6 x 3  10 x 2  ax  b is a perfect square, find the value of a and b.
19.
i)
ii)
The polynomial p( x)  x 4  4 x 3  bx 2  cx  d is a perfect square of second
20.
If f (x ) is a polynomial in x , show that when f (x ) is divided by ( x  a ) the remainder is
f (a ). When x 3  ax 2  bx  c is divided by ( x  3) the remainder is -26 and when divided
by x 2  x  2 the remainder is 14. Find the values of a, b, c.
21.
If the polynomial function of second degree in y leaves remainders 1, 25, 1 on division by
y  1, y  1, y  2 respectively. Show that the function is a perfect square.
22.
If ( x  1) 2 is a factor of 2 x 4  7 x 3  6 x 2  Ax  B and has remainder 14 when divided by
x  1 , find A and B .