MIPU3 H2 Mathematics Paper 2 2012 Question

2 Section A: Pure Mathematics [40 marks] 1 (a) The complex number z satisfies 3 4 5zi   . (i) On an Argand diagram, sketch and shade the region in which the point representing z can lie. [3] (ii) Find the range of possible values of 1z . [2] (b) Using a single Argand diagram, sketch the loci given by (i) 21zi , (ii)   1 4arg zi  . [3] Hence, find the value of z, in cartesian coordinate form x iy , that satisfy both (i) and (ii). [1] 2 The function f is defined by 2:1 xf x e  , x . (i) Show that 1f  exists and state its range. (ii) Find 1f  and write down its domain. (iii) Sketch the graph of 1f  , giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the axes. (iv) Hence, find the value(s) of x that satisfies the equation 2 1xex  . [2] [3] [3] [3]

3 3 The curve C has parametric equations 2 1xt , 1 2y t  , 2t . (i) Find dy dx in terms of t. [3] (ii) Find the equation of the normal to curve C at point P (2, -1). (iii) Find the values of t for the other points of intersection between the curve C and the normal found in part (ii). [4] [3] 4 (a) Prove that     1 6 1 4 1 2 11 n r r r n n n      . [5] (b) (i) Prove by the method of differences that  1 11 111 n r r r n  . [3] (ii) Explain why  1 2 1r rr    is a convergent series, and find the value of the sum to infinity. [2]

4 Section B: Statistics [60 marks] 5 (a) An interior designer is designing a tile pattern of 3 blue, 3 red, 3 green, 1 yellow and 1 purple tiles in a line. All the tiles are identical except for the colour. Find the number of different possible ways to arrange the tiles if (i) the green, yellow and purple tiles must be placed as a single block, (ii) no blue tiles are placed next to each other, (iii) a red tile to be placed at the beginning and at the end of the line. [3] [3] [3] (b) A group of 10 people consists of 9 men and 1 woman. Find the number of ways which the group can be seated at a round table with identical chairs if (i) there is no restriction, (iii) 2 particular men, Caleb and James, do not want to sit beside the woman, but will like to sit together. The chairs at the table are replaced with 10 chairs of different colours. (iii) Find the number of ways which the group can be seated at a round table if Caleb and James must still sit together, but they need not be separated from the woman. [1] [3] [3] 6 There are 4 white and 5 green balls placed in a bag and the player picks 2 balls from the bag without replacement. The player gets to roll a fair die once if both balls are of the same colour, and twice if both balls are of different colours. The player wins a DVD every tim