RVHS P2

Name ( ) Class RIVER VALLEY HIGH SCHOOL 2012 Year 6 Preliminary Examination Higher 2 MATHEMATICS Paper 2 Additional Materials: Answer Paper List of Formulae (MF15) Cover Page 9740/02 14 September 2012 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number in the space at the top of this page. Write your name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, place the cover page on top of your answer paper and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

Section A: Pure Mathematics [40 marks] 1 Use the substitution u xy , where u is a function of x, to reduce the differential equation 32d (1 ) 0d yx x y xx    to 2d ( 1)d u uxx  . Hence show that the general solution to the differential equation is of the form f ( )e1 xAy x  , where f is a function in x. [5] Sketch on separate diagrams a member of the family of curves corresponding to (i) 1A and (ii) 1A . [2] 2 (a) Use the substitution tan e xy to find 2 1 de (1 e )xx x . [4] (b) The region bounded by the x-axis, the curve 2 1exy  , the lines 0x and 1x is rotated about the y-axis to form a solid of revolution of volume V. Find the exact value of V. [6] 3 (a) The diagram shows the graph of  fyx . It has a maximum point at at  ,2a where 40 a   and passes through the points  6, 0 and  2, 0 . The curve has asymptotes at 0x and 4y  . On separate diagrams, sketch the graphs of (i)  f'yx , [3] (ii) 1 f ( )y x , [3] (iii)  f4yx [3] (b) Describe a s equence of transformations that maps the graph of 2 2 14 yx  onto the graph of   22 11xy   . [2] 6 O x y 2  ,2a 4y 

4 (i) Solve the equation 3 80z  , giving the roots in the form ier  , where