page 05 MARKS 7. (a) Find an expression for   2 1 3 n r r r    in terms of n. Express your answer in the form    1 3 n n a n b   . (b) Hence, or otherwise, find   20 2 11 3 r r r    . 8. (a) Consider the statement: For all integers a and b, if a < b then a2 < b2. Find a counterexample to show that the statement is false. (b) Let n be an odd integer. Prove directly that n2 − 1 is divisible by 4. 9. (a) State the matrix A, associated with an anti‑clockwise rotation of 2 π radians about the origin. The matrix B is given by 3 1 2 2 3 1 2 2 B                 The matrix given by AB is associated with an anti-clockwise rotation of α radians about the origin. (b) (i) Determine AB. (ii) Find the value of α. (c) Determine the least positive integer value of n such that (AB)n = I, where I is the 2 × 2 identity matrix. [END OF QUESTION PAPER] 2 2 1 2 1 1 1 1 page 06 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 07 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE