ASRJC_FM_2023_Prelim_P2 (Questions)

FURTHER MATHEMATICS 9649/02 Paper 2 18 September 2023 3 hours Additional Materials: Answer Booklets 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 booklet. If you need additional answer booklet, ask the invigilator for a continuation booklet. Write your name and class on the cover page and on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page. ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2023 Higher 2

2 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 1 The points P and 'P in the Argand diagram represent the complex numbers z and 'z respectively. The transformation : ',PPT where ( ) ',PPT is given by 2 *' kz z where k is a non-zero real constant and *z is the conjugate of z. Show that the line 'PP passes through the origin O. [2] Let Q represent the complex number w and ' ( ).QQ T If P and Q lie on the circles with centers at the origin and of radius 2 and 3 respectively , find the value of k for which the ratio of the area of triangle ''OP Q to area of triangle OPQ is 4 :1. Give your answer in exact form. [3] 2 Let nF be the sequence of Fibonacci numbers : 1 2 1 21, 1 and for all , 3n n nF F F F F n n       and nL be the sequence of Lucas numbers: 1 2 1 21, 3 and for all , 3n n nL L L L L n n       It can be proven that 11 for all , 1n n nL F F n n    . (a) List down the first 6 Fibonacci numbers and Lucas numbers and verify that 11 is true for 5n n nL F F n   . [2] (b) For a fixed m where 2m , prove by induction on n that 11m n m n m nF F F F F   . [5] (c) Hence or otherwise, show that 2n n nF F L . [2] 3 (a) Show that   2 21 4 sin . 2 iiee  