AJC_H2_MATH_P1_Question

Page 1 of 4 AJC / 2015 Preliminary Examination / 9740 / P1 Anderson Junior College 2015 Preliminary Examination H2 Mathematics Paper 1 (9740/01) 1 Without using a graphic calculator, solve the inequality 2 2121 x xx    . [3] Hence find the exact solution of the inequality 22 22 21 2 2 x x x x    . [3] 2 It is given that 2 d5 d yx x y  and when 0, 5.xy  Find the Maclaurin’s series expansion for y, up to and includi ng the term in 3,x leaving the coefficients in exact form. [6] 3 The complex number z satisfies 43 i 2z  and 2arg ( 4) 2z   (i) Sketch clearly the locus of z on an Argand diagram. [3] (ii) Find the range of values of | 8 |z . [2] (iii) Find maximum value of arg( 8)z . [3] 4 The diagram shows the curve C with equation 12siny x and the line L with equation 8 3yx   . C and L intersect at the point where 1 2x . The region S is defined by 12sinyx  , 8 3yx   and 0x . Find the exact volume of the solid obtained when S is rotated through 2 radians about the y-axis. [6] L C y x

Page 2 of 4 AJC / 2015 Preliminary Examination / 9740 / P1 5 In the triangle ABC, angle radians,BAC  angle radians6ACB  and 3.AC  Given that θ is sufficiently small, show that 2 2 23 , 22 3 AB a b c        where a, b and c are constants to be determined in exact form. [7] 6 Do not use a graphic calculator in answering this question. It is given that 2sin xx  for 0 2x   . (i) Explain why 2 22 sin 00 ed e d x x x x     . [2] (ii) By making the substitution ux   , show that 2sin sin 02 ed edxu x u     . [2] (iii) Hence show that  sin 0 ed e 1 e x x     . [3] 7 A curve C1 has the equation 22 2 2p xy p  where 1p . (i) Sketch C1, stating the coordinates of any poi nts of intersection with the axes, the coordinates of any stationary points and the equations of any asymptotes in terms of p. [3] (ii) C1 undergoes a single transformation to become C2. Given that C2 has a line of symmetry 2x and the point (4,3) lies on C2, find p. [2] (iii) The graph of f ( )y x is given below. It has a maximum point at 1x and a horizontal asymptote 0y . Sketch the graph of f( )yx  on the same diagram as C1. Hence, state the number of roots of the equation   222 2 f()p xx p   . [4] y x 1

Page 3 of 4 AJC / 2015 Preliminary Examination / 9740 / P1 8 (i) Prove by the method of mathematical induction that  2 21 1 2 9 (3 ) (5 ) 3 0 4 5 n r n rr n n    . [5] (ii) Hence find 4 4 2 (2 ) n r rr    . [3] (iii) Deduce that 4 2 4 19 40(1 ) n r r     . [ 2 ] 9 Show that for x > 1, (i)  32 2 2 d1 d 1 1 x x x x     . [1] (ii) 1 2 d1 1sind 1xx xx    [2] A curve C has parametric equations