2019 J2 JPJC H2 Math Prelim (with ans)

JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2019 MATHEMATICS 9758/01 Higher 2 18 September 2019 Paper 1 3 hours 1( i) The first four terms of a sequence are given by 1 13u  , 2 12.8u  , 3 1.8u  and 4 38u  . Given that mu is a cubic polynomial in m, find mu in terms of m. [3] (ii) Find the range of values of m for which mu is greater than 2000. [2] 2( a) Express 31 2 xy x   i n t h e f o r m 2 ByA x  , where A and B are constants to be found. Hence, state a sequence of transformations that transforms the graph of 1y x to the graph of 31 2 xy x   .[ 4] (b) It is given that 2g( ) 2 2xx x . Sketch the graph of g( )yx , stating clearly the coordinates of any turning point s and axial intercepts. Find numerically, the volume of revolution when the region bounded by the curve g( )yx and the line 5y  is rotated completely about the x-axis. [5] 3 Referred to an origin O, the points A and B are such that OA  a  and OB  b  . The point C is such that OACB is a parallelogram. The point D is on BC such that BD BC     and the point E is on AC such that AE AC     , where  and  are positive constants. The area of triangle ODE is k times the area of triangle OCE. (i) By finding the area of triangle ODE and OCE in terms of a and b, find k in terms of  and  .[ 6] (ii) The point F is on OC and ED such that :6 :1OF FC  and :3 : 4DF FE  . By finding the values of  and  , calculate the value of k.[ 3] 4 The curve C has equation 2 5 2 xy x   . (i) Prove, using an algebraic method, that C cannot lie between two values to be determined. [4] (ii) Sketch C, showing clearly the equations o f any asymptotes and coordinat es of any turning points and axial intercepts. [4] (iii) By adding a suitable graph to your sketch in (ii), deduce the range of values of h for which the equation      2 22 222 51 2 2xx x h x     has at least one positive real root. [3] [Turn over www.KiasuExamPaper.com 342

2 5 Do not use a graphing calculato r in answering this question. (a) (i) It is given that 1 35 iw   . Find the value of 3 1w , showing clearly how you obtain your answer. [2] (ii) Given that 35 i is a root of the equation 3241 4 0wp wq w   , using your result in (i) , find the values of the real numbers p and q. [3] (iii) For these values of p and q, find the other two roots of the equation in part(ii). [3] (b) It is given that 13 iz   . Find the set of values of n for which * n z z is purely imaginary. [4] 6 The function f is defined for all real x by  22fe 9 exxx  . (i) Show that f ' 0x  for all x. [2] (ii) Show that the set of values of x for which the graph fyx is concave upward is the same as the set of values of x for which f0 x  , and find this