RI_DHS_HCI_TMJC_9649_2023_Prelim_P1

This document consists of 8 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 [Turn over FURTHER MATHEMATICS 9649/01 Paper 1 September 2023 3 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet and a graph paper booklet will be provided with this question paper. You should follow the instructions on the front cover of both booklets. If you need additional answer paper or graph paper ask the invigilator for a continuation booklet or graph paper booklet. 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 an 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. RAFFLES INSTITUTION 2023 YEAR 6 PRELIMINARY EXAMINATION

2 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 1 1 (a) Show, without sketching a graph, that there is exactly one root,  , of the equation 32 e 2 0xx    in the interval [1, 2]. [2] (b) Use linear interpolation once, on the interval [1, 2] , to find an approximation, 1 , to  , giving one decimal place in your answer. [2] (c) Use the Newton-Raphson method, with the starting value 1 , to find the value of  correct to 3 decimal places. [2] 2 A sequence of real numbers 1,u 2,u 3,u satisfies the recurrence relation 1ua and 1 8 21 2 n n n uu u    for 2a and 1n . (a) Prove algebraically that, if the sequence converges, then it converges to either 3 or 7. [2] (b) Use an algebraic method to prove that 1 7nnuu  if 37 nu . [4] (c) Describe the behaviour of the sequence for 1 4u  . [2] 3 The terms in the sequence 0x , 1x , 2x , ... satisfy the recurrence relation 2122 n n nax x a x where a is a positive constant and 0n . (a) Find the general solution of this recurrence relation. [3] (b) It is given that 3a , 0 3x  and 1 1 3 x  . Show that    f ( )