RI 2023 JC1 Promos 9649

This document consists of 6 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 [Turn over FURTHER MATHEMATICS 9649 September 2023 3 hours Additional materials: List of Formulae (MF26) Writing Paper READ THESE INSTRUCTIONS FIRST 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 5 PROMOTION EXAMINATION

2 H2 FM 9649/2023 RI Year 5 Promotion Examination 1 A sequence 012,,, . . .uuu is such that  56An n nuB where A and B are constants and 0n  . Given that 1 79u  and 2 949u  , find the values of A and B. [3] Prove by induction that  410 2 3 nn nu  is a multiple of 19 for all non-negative integers n. [ 5 ] 2 The equation of an ellipse is 22 22 1xy ab  where 1F and 2F are its foci. 1F is at  ,0c , 2F is at  ,0c where 0ac and 22 .ba c (a) Given that the point  ,P xy lies on the ellipse, show that 2 1 ac xPF a  and hence 12 2.PFP F a [4] (b) Given that a point  4, 3Q is on the ellipse and 12 ,QF QF k show that 2 2 Aka B aB    where A and B are constants to be determined. [4] 3 (i) Using the recurrence relation 1 1 ln( 1)3 nnxx  with 1 0.8x  , write down 2x , 5x and 8x . Determine which root of the equation 3l n (1 ) 0xx  does the sequence converge to. [2] It is given that f( ) 3 l n ( 1 )xx x    and  is very close to zero. It is known, from graphical work, that the equation f( ) 0x  has two single roots, x  and x  , where   . (ii) Explain why  is close to 0. [1] (iii) The Newton-Raphson iterative formula can be written as  1 gnn nx xx  , where     fg f' xx x . Write down g x in this case. [1] (iv) The Newton-Raphson method is used to find the value of . Use two iterations of the Newton-Raphson method,