RVHS_H2_MATHS_P2_QP

1 2015 RVHS H2 Maths Prelim Paper 2 Section A: Pure Mathematics [40 marks] 1 The curve C has equation bx axxy   12 . (i) Given that the graph passes through the point  0,1 and has a vertical asymptote at 2x , find the values for a and b. [2] (ii) Sketch the graph of C, clearly stating the equations of any asymptotes and the coordinates of any axial intercepts and/or turning points. [3] (iii) Hence, find the range of values of k for which the equation     21 2  xkxx has two real roots, where k . [3] 2 The functions f and g are defined by: 1 ,1 , 1 12:g 1 ,34:f 2 2    x x x xxxx   (i) Show that f is one-one. [1] (ii) Find f-1 in a similar form. [3] (iii) Find the largest domain of g such that fg exists. [2] (iv) Hence find the range of fg. [3] 3 The complex numbers z and w are such that | 2i | 1 arg( ) z wk   (i) Given π 4k  , sketch the loci on an Argand diagram. Give the geometrical description of the loci. [4] (ii) Find the values of k such that the loci intersect at exactly one point. [3] (iii) Given that 5π 6k  , find the values of w and z in the form x + iy that minimize |w – z|. [4]

2 2015 RVHS H2 Maths Prelim Paper 2 4 (a) Show that the differential equation 2d 4 ( ) 0d ux u uxx     may be reduced by means of the substitution 2y ux to 2d 4d y yyx  . [2] Hence, find the general solution of the differential equation , leaving your answer in exact form. [4] (b) The displacement s (metres) of an object moving in a straight line from a fixed point O is related to time t (seconds) by the differential equation 2d 4d s sst  . (i) Sketch the solution curve of the particular solution for 04 πt given that 1s when 5π 6t  . [4] (ii) Describe the motion of the object and comment on whether the differential equation in s and t is an appropriate model in the real-life context. [2] Section B: Statistics [60 marks] 5 A committee consists of i representatives from faculty I, j representatives from faculty J and k representatives from faculty K. Find the number of ways of arranging the members of the committee (i) in a circle for a meeting; [1] (ii) in a row with representatives from the same faculty seated together. [2] 6 (a) Explain briefly the difference between a sample and a population. [2] (b) A school has 120 teachers belonging to the following four departments: Humanities 21 Language 33 Mathematics 27 Science 39 The Principal wants to gather feedback from the staff regarding some school programmes. A sample of 40 teachers from the four departments is to be chosen to take part in a survey. (i) The sample is formed by randomly selecting 10 teachers from each of the four departments. Explain why this procedure will not result in a random sample of teachers. [2] (ii) Describe how a stratified random sample of the 40 teachers may be obtained.[2]

3 2015 RVHS H2 Maths Prelim Paper