2020 J2 VJC H2 Math prelims (Paper 2)

[Turn Over Section A: Pure Mathematics [43 m] 1 The shaded region in the diagram above shows the cross sectional view of a cylinder open on both ends inscribed in a hemisphere with fixed radius n cm. If the diameter of the cylinder is x cm, show that the surface area A cm2 of the cylinder is given by 222A x n x . [2] Use differentiation to find, in terms of n, the value of x which gives a maximum value of A, justifying that this value gives maximum A. Find also the ratio of the diameter of the cylinder to the height of the cylinder, in this case, simplifying your answer. [5] 2 The points A and B have positions vectors 2i j k and 4sti j k respectively. The plane p has equation 3 2 5x y z   . Given that the line AB has no points in common with the plane p, express s in terms of t. [2] Find the shortest distance between the line AB and the plane p. [2] Another plane q is perpendicular to plane p and c ontains the line with equation 5 1 1 3 5 2 x y z   . Find the equation of plane q in cartesian form. [3] Find the position vector of the point which is a reflection of A in the line of intersection between p and q. [6] 3 (a) The complex numbers w and z are such that 1 i 3w  and *0zz  . The points P and Q in the Argand diagram represent the complex numbers w and *zz  respectively. Find  arg w , leaving your answer in exact form. Hence, state the possible values of angle POQ, where O is the origin. [3] (b) Given that 1z and 2z are complex numbers such that 12 12 3 3 zz zz   is purely imaginary, show that 21 3zz  . [3] 4 (a) (i) Find the derivative of 2 2exx . [1] (ii) Hence, find   23 21 e dxxxx  . [3] (b) By using the substitution sinxa  , find 22 0 d a a x x . [4] (c) (i) On the same diagram, sketch the graphs of 2 2yx and yx for 0 2.x [2] The region R is bounded by the curve 2 2yx , the line yx and the x-axis. (ii) Find the exact area of R. [4]

2 (iii) Find the exact volume of the solid obtained when region R is rotated through 4 right angles about the y-axis. [3] Section B: Statistics [57 m] 5 For events A and B, it is given that  P 0.6A  ,  P 0.7AB  and  P | 0.3BA  . (i) Show that  P 0.28B  and determine with a reason, if A and B are independent.[3] (ii) An event C is such that A and C are mutually exclusive and  P 0.25C  . Find the range of values of  P' BC . [3] 6 A group of 8 people consists of 4 boys and 4 girls. (a) They go to a theatre and sit in the last row of 10 adjacent seats at random. (i) Find the number of ways in which they can be seated if they are not all seated together. [2] (ii) Given that they are not all seated together, find the probability that the boys are seated together and the girls are also seated together. [2] (b) At a dinner, the group si