NYJC TJC VJC FM 2024 Prelim P2

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 12 September 2024 3 hours Additional Materials: Answer Booklet 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 answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Write your Civics Group and name on all the work that 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. 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. [Turn over

2 Section A: Pure Mathematics [50 marks] 1 (a) Solve the equation 3 32 32 3iz =− , giving your answers 1z , 2z and 3z in exact form ie,r  where 0r ,   −   and 1 2 3arg arg argz z z . [3] (b) The roots 1z , 2z and 3z in part (a) are represented by points 1P , 2P and 3P . Find the exact perimeter of triangle 1 2 3PP P . [2] 2 The Fibonacci numbers nF are defined by the conditions 0 0F = , 1 1F = and 11n n nF F F+−=+ for all 1n . (a) Compute 23,FF , 4F and 5F . [1] (b) Compute 2 11n n nF F F+− − for 1, 2,3, 4n= . [2] (c) Conjecture and prove by induction, for all 1n , an expression for 2 11n n nF F F+− − . [6] 3 At the start of the month in January 2023, Sandy started her career as a lifestyle vlog content creator with 60 subscribers for her channel. By the end of each month, she lost 5% of her existing subscribers but gained 100 new subscribers each month through her publicity efforts. Let nu be the number of subscribers at the end of the nth month taking January 2023 as the first month. (a) Write down a recurrence relation to model the number of subscribers at the end of the nth month. [2] (b) Find the general formula of nu in terms of n. Hence show that the number of subscribers at the end of December 2023 is 952, correct to the nearest integer. [4] (c) Based on the model in (a), comment on the long-term prospect of Sandy’s career.