Page 05 MARKS 6. Find Maclaurin expansions for sin 3x  and  e4x up to and including the term in x3. Hence obtain an expansion for e4x sin 3x up to and including the term in x3. 7. A is the matrix 2 0 1 λ ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ . (a) Find the determinant of matrix A. (b) Show that A2 can be expressed in the form pA + qI , stating the values of p and q. (c) Obtain a similar expression for A4. 8. Let 3 z i = −. (a) Plot z on an Argand diagram. (b) Let w = az where a > 0 , a ∈ . Express w in polar form. (c) Express w8 in the form ( ) n ka x i y + where k, x, y ∈. 9. Obtain ( ) ln 2 7x x dx ∫ . 10. For each of the following statements, decide whether it is true or false. If true, give a proof; if false, give a counterexample. A. If a positive integer p is prime, then so is 2p + 1. B. If a positive integer n has remainder 1 when divided by 3, then n3 also has remainder 1 when divided by 3. 11. The height of a cube is increasing at the rate of 5 cm s-1. Find the rate of increase of the volume when the height of the cube is 3 cm. [Turn over 6 1 3 2 1 2 3 6 4 4