2012_TJC_JC2_H2_Prelim_Paper_2

1 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination Higher 2 MATHEMATICS 9740/02 Paper 2 19 September 2012 Additional Materials: Answer paper 3 hours List of Formula (MF15) READ THESE INSTRUCTIONS FIRST Write your Name and Civics 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 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 mathematic al 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 7 printed pages. [Turn Over

2 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 Section A: Pure Mathematics [40 marks] 1 A curve C is defined by the parametric equations 2 3 cos 3x  ,  2 3 1 siny  where 02 π. (i) Find d d y x in terms of  . [2] (ii) Show that the Cartesian equation of the curve C is 2 2 3 1 2 3 2 3 1 xy     . Hence or otherwise, sketch the graph of C, indicating clearly the x-intercepts in exact form. [3] (iii) The point P 32 3, 3 2   lies on curve C. The region R is bounded by the curve C for 3x , the x-axis and the line segment joining the points P and  3,0 . Show that the area of R is     3 2 0 3 2 3 1 2 6 3 sin d4      units2. [4] 2 The diagram above shows the curve of 2 ex xy . Two points A and B on the curve have coordinates (, 1 2 ) and (, 1 2 ) respectively. A sequence of real numbers 1 2 3, , ,...x x x satisfies the recurrence relation 1 1 e4 nx nx   for n  1. (i) Show algebraically that if the sequence converges, then it converges to either  or . [3] (ii) Show that 1nx   if nx <  . [2] (iii) Show that 1nnxx  if nx <  . [2] (iv) Explain briefly how the results in (ii) and (iii) may be used to deduce that the sequence converges to  when 1 0x  . [2]

3 © TJC 2012 TJC/MA 9740/Preliminary