JPJC_9649_2023_Prelim_P2

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2023 FURTHER MATHEMATICS 9649/02 Higher 2 18 September 2023 Paper 2 3 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklets. If you need any additional answer paper ask the invigilator for a continuation booklet. 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupport ed answers from a graphing calculator are not allowed in a question, you are required to pre sent the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 8 printed pages. [Turn over

2 Section A: Pure Mathematics [50 marks] 1 The sequence  nX is given by   2 1 2 n n n XX X    , for 1n . By applying th e natural logarithm to the recurrence relation, use a suitable substitution to find the general solution of the sequence, expressing your answer in trigonometric form. [5] 2 Let   1 2 1 1d n nI x x   , for 22 x   . (a) Use integration by parts to show that   12 1 2 nnn I n I  , for 0n . [3] (b) Hence, or otherwise, evaluate 3 2 I , giving your answer as a multiple of  . [3] (c) Use Simpson’s ru le with five ordinates to find a n approximation to 3 2 I , giving your answer in exact form. [2] (d) Deduce an approximation to  , giving your answer in exact form. [1]

3 3 The diagram above shows a circle with radius r cm. Initially, the lowest point P on the circumference of the circle is at the origin. As the circle rolls along the x-axis in the positive direction, P traces a path and this locus is known as a cycloid. At any time, the angle through which the rolling circle has rotated is  radians. Show that the locus of P has parametric equations    sin , 1 cosx r y r      . [2] The part of this locus for which 02  is denoted by C. (a) Sketch C and find the coordinates of its maximum point in terms of r. [3] (b) Show that the su rface area formed when C is completely rotated about the x-axis can be expressed as the integral 223 0 8 sin d 2r    , and hence, find its exact area in