ASRJC_9649_2025_Prelim_P1

[Turn over FURTHER MATHEMATICS 9649/01 Paper 1 27 August 2025 (Wednesday), 2 – 5 pm 3 hours Additional Materials: Pr inted 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 5 printed pages and 3 blank pages. Anderson Serangoon Junior College ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2025 Higher 2

2 Anderson Serangoon Junior College 1 A surface has equation  f,zx y where 2x uv and 2y uv . (a) Show that ff 1 2uv   if the tangent plane at the point P(a, b, c) is parallel to the plane with equation 1x z where a > 0. [3] (b) Given that x yzxy x y   , find the exact coordinates of the point P. [4] 2 (a) V denotes the set of vectors of the form , a b c    where a, b and c are real numbers. (i) Show that the subset 22: a bV ab c    does not form a linear space. [1] (ii) Determine whether :2 0 a bVa b c c      forms a linear space, justifying your answer. [3] (b) It is given that the square matrix M satisfies 2 23 MM I 0 . Deduce that 3 6MMI . Hence show that 876 24MMM M can be expressed as (3 )k MI , where k is a constant to be determined. [4] 3 By using the substitution d 2d ywx x , show that the general solution of the differential equation  32 2 32 ddd 44 8 1 eddd xyy y xxxx     is  22 2 12 1 e28 x x Cx C x D    , where 1C , 2C and D are arbitrary constants. [9]

3 Anderson Serangoon Junior College [Turn over 4 Consider the second-order linear recurrence relation 21 71 0 0nn naa a  , where 0n , 01 2, 9aa . (a) Expr