HCI_JC_2_H2_Maths_2012_Paper 2 Questions

© Hwa Chong Institution 2012 9740/02/Prelim/12 [Turn over HWA CHONG INSTITUTION Preliminary Examination Higher 2 MATHEMATICS 9740/02 Paper 2 14 September 2012 3 hours Additional Materials: Answer 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 calculator 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 6 printed pages.

2 © Hwa Chong Institution 2012 9740/02/Prelim/12 Section A: Pure Mathematics [40 marks] 1 The functions f, g and h are defined as follows: f : sin( )xx  , , 0 2xx   , g: xxa , ,1xx  , where a is a constant and 01 a , h : f ( )xx , ,x m x n   . (i) Explain why 1f  does not exist. Find m and n so that 1h exists and the range of h is equal to the range of f. Find 1 1h 2    . [3] (ii) Show that the composite function gh exists, and find the range of gh in terms of a . [3] 2 A Hwa Chong student taking part in Project Day needs to design a rectangular poster to showcase his project. According to the rules set for the poster design, the poster should contain 1352 cm2 of printing with margins of 4 cm each at the top and bottom, and 2 cm each on the left and right side s. By using differentiation, determine the dimensions of the poster to minimise the amount of paper used. [6] 3 (i) Show that for any complex number ie,z  1 2cos .n nzn z  [2] (ii) By taking 1,n show that 33 3 1 3 1cos 3 . 8 zz zz     Deduce that 3 13cos cos3 cos .44   [4] (iii) Find 3cos 3 d . [2]

3 © Hwa Chong Institution 2012 9740/02/Prelim/12 [Turn over 4 (a) Referred to the origin O, the position vectors of three