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 2 1 1 x t   , lny t where t > 1. R is the finite region bounded by the curve C, the x-axis and the lines 1 3 x and 3x . Show that the area of R is given by   2 32 2 23 ln d 1 tt t t  . By using the results in (i) & (ii), find the exact area of R. [6] 10 The function f is defined by 2 1f : 2 x x  , ,0 , 2xx x   (i) Define the inverse function 1f  in a similar form. [3] (ii) Sketch the graphs of f and 1f  on the same diagram, giving the exact equation of any asymptote(s) and showing clear ly the relationship between the two graphs. Hence find the set of values of x , in exact form, for which    1ff x x . [6] Another function g is defined by g : 1 e xx  , , 0xx  where  is a constant. (iii) Given that the composite function 1fg exists, find the greatest value of . [2] With this value of  , find the range of 1fg . [1]

Page 4 of 4 AJC / 2015 Preliminary Examination / 9740 / P1 11 A piece of vanguard sheet, ABCDEF, is in the form of a regular hexagon of side a cm. A kite shape is cut out from each corner to form the shaded shape, as shown in Fig. 1. It is then folded to form the open hexagonal box of height h cm, as shown in Fig. 2. (i) Show that the volume V of the box in cm3 is given by 2 323 2Vh a h   . [4] (ii) Use differentiation to find, in terms of a, the value of h which would result in the volume of the box being maximum. [4] 12 The line l passes through the points (0, 0,1)P and (0, 6, 2).Q The plane p1 is perpendicular to the vector 2 i j k and contains the point Q. (i) Find the acute angle between line l and plane p1 . [3] (ii) The point R lies on the x-y plane such that its distance from the mid-point M of the line segment PQ is 2 units. If MR is perpendicular to the line l, find the coordinates of R. [5] The plane p2 is given by the equation 5 10 10 0xy z   . (iii) The plane p3 passes through the point (1,1,1 ) and contains all the common points of p1 and p2. Find a vector equation of p3, giving your answer in the scalar product form. [4] END OF PAPER A B C D E F a cm Fig. 1 Fig. 2 h cm