2a Let              

  1    2   −1
A   =        4  k    −2
0    0   −1

−k      2     k − 4
B =       4     −1       −2
0       0     k − 8

 
i    Express AB in the simplest form
ii     Find the inverse of A in terms of k, stating the condition under which A−1 exist
x  0 
iii     Given that X = y and C =  46 , k=5 use the inverse of A to solve the equation AX=C

2b Given that a and b are positive, solve the simultaneous equation
a = 3b log3 a + log3 b = 2
10 marks 2c P (x) = x3 + x2 + ax + b. If P(x) is divided by (x                                        2), the remainder is                                              7 and If P(x) is divided by
(x + 1), the remainder is 32. Find the value of a and b
3a A transformation is defined by the 2 × 2 matrix
A   = a b a a + b a
 
where a and b are scalars. If n is an odd integer, prove that
An     =  bn−1A​