ASRJC_9649_2025_Prelim_P2

[Turn over FURTHER MATHEMATICS 9649/02 Paper 2 29 August 2025 (Friday), 8 – 11 am 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your name on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE ON ANY BARCODES. Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages. Anderson Serangoon Junior College ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2025 Higher 2

2 Anderson Serangoon Junior College Section A: Pure Mathematics [50 marks] 1 �e Fibonacci numbers is defined by the recurrence relation 21nnnUUU−− += where 2n≥ and 01 1UU= = . { }nR is a sequence of real numbers where 1 n n n UR U − = , n +∈  . Given that this sequence converges, prove algebraically that its limit can be expressed in the form 2 ab+ , where a and b are integers to be determined. [5] 2 (a) As an alternative to the Newton-Raphson method for numerical root-finding, Chebyshev’s method computes successive approximations using the following recurrence relation: [ ] 2 1 3 f () f () f () f( ) 2f( ) n nn nn n n x xxxx x x + ′′= −− ′ ′ , n +∈  . Determine the general form that f( x) must take such that Newton- Raphson method and the Chebyshev’s method will lead to the same subsequent approximation given any initial value in the domain of f. [2] �e equation ln 0xx = has a root 1α = . (b ) Using a common initial approximation of 1 5x = , perform successive iterations of both Newton- Raphson and Chebyshev’s methods, and comment on your observations. [4] (c) Explain why Chebyshev’s method fails with an initial approximation of 0.5. [2] 3 �e Himmelblau function is defined by ( ) ( ) ( ) 2222f , 11 7xy x y x y= +− + + − . (a) Find the unit vector in the direction of the steepest descent at