2012_VJC_H2_MA_Prelim_P2_Questions

1 VICTORIA JUNIOR COLLEGE Preliminary Examination Higher 2 MATHEMATICS 9740/02 PAPER 2 September 2012 3 hours Additional materials: Answer paper Graph paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and CT group 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. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. 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 calcul ator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, 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 5 printed pages © VJC 2012 VICTORIA JUNIOR COLLEGE [Turn over

2 Section A: Pure Mathematics [40 marks] 1 The complex number z satisfies  arg 1 2iz    , where  is a fixed angle in the int erval      . (i) Give a geometrical description of the locus of the point P representing z. [1] (ii) Given that π 6  , find the exact minimum value of 1 6iz . [3] The complex number w satisfies 6 2i 2(cos isin )w     , where      . Sketch the locus of the point Q representing w. [2] [2] (iii) Find the range of values of  such that the locus of P meets the locus of Q more than once. [2] 2 A sequence 0u , 1u , 2u , … is such that 0 1u  and   1 12 1 nnu u n nn     , for all n  . (i) Express   1 1rr  in partial fractions. Hence find  1 12 1 n r r rr     in terms of n. [4] (ii) By considering 1rruu  and using the result in part (i), show that 1 ( 1) 2 1 nu n n n     , where n  . [3] (iii) Using the result in part (i), find   1 1 12( 1) ( 1) 2 n r r rr       in terms of n. [3] 3 The above diagram shows the curve C with equation 22e y xy for x > 0. (i) Make a copy of the diagram to include the portion of C for x < 0. [1] (ii) Show that 2 d2 d 2e 1 y yx x   . [2] (iii) Given that the line xk is a tangent to C, find the possible exact values of k. [4] (iv) Find the equation of the tangent to C at the point  1,0 and determine the x-coordinate of the point where this tangent meets C again. [4] x y o

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