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