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) Express this relation in matrix form 1nn vA v , where 1 ,0 n n n a na   v . [1] (b) Without using a calculator, find the eigenvalues of A and the corresponding eigenvectors. [4] (c) By writing A in the form 1PDP , find na in terms of n. [3] (d) Explain whether the approach used in (d) would work for all second-order linear recurrence relations of the form 21 0nnnaaa     , , /Rbb. [1] 5 (a) Show that 1 1 0 1e 1 dt a nee e 1xx x       . [3] (b) Two students want to obtain numerical approximations of 1 0 1 deexx x  . Student A uses Simpson’s rule with a step-size of 1 6 , while Student B uses standard series (in MF27) to obtain the series expansion of 1 eex x , up to the 2x term, followed by integration. Find the percentage errors incurred for each of the two methods. [6] (c) ๠e two students now want to obtain the numerical approximation of 3 0 1 deexx x  using their previously chosen methods. Without performing further computations, suggest with reason which student would obtain a better approximation. [1] 6 (a) By considering the series 23 nzz z z … , where cos i sinz   , show that (1 )cos cos 2 cos3 cos sin cos cosec 22 2 nnn      … , and obtain a similar expression for sin sin 2 sin 3 sin n    … . [5] (b) Obtain a simpli fied expression for 22 11 sin cos nn rr rr        in terms of n and θ. [1] (c) Show that, for all positive integers n, the expression 22 24 2 2 4 2sin sin sin cos cos cos11 1 1 1 1 nn nn n n n n                   …… is equal to a constant independent of n. [2]

4 Anderson Serangoon Junior College 7 (a) Show that for positive integers 2n , 1 21c o s s i nsin d 1 sin d n nn x xxx xx nn    . [3] (b) A polar curve 1C has equation 4sin , 0r    . ( i ) Find the exact area enclosed by 1C . [3] Another polar curve 2C has equation 4cot , 0r    . (ii) Show that the y-coordinate of the points of intersection of the two curves is 5 2  , where 15 2  . [3] (iii) Find the length of 2C that is enclosed within 1C , giving your answer correct to 3 decimal places. [2] 8 Consider a curve segment satisfying  21 2l n4yx x , with endpoints (2, 1 ln 2) and (4, 4 ln 2) . (a) Show that the length of the curve segment is 3l n2 . [3] (b) Find the exact area of the surface generated wh en the curve segment is rotated fully about the x- axis, simplifying your answer. [4] (c) Find the volume of the solid formed when the region bounded by the curve segment, the y-axis and the horizontal lines 1y  and 3y is rotated fully about the y-axis. [4] 9 ๠e function ()yy x satisfies d cos sin cosd y yx xxx . ๠e value of ()yh is to be found where h is a small positive number such that terms in 3h and above are negligible, and 0(0)yy  . (a) Use two steps of Euler’s method to determine an approximation to ()yh in terms of h and 0y . [3] (b) Use one step of the improved Euler’s method to determine an alternative approximation to ()yh in terms of h and 0y . [2] (c) Solve the di fferential equation analytically and find ()yh in terms of h and 0y . [7] (d) Comment on the results obtained in (a), (b) and (c). [1]

5 Anderson Serangoon Junior College [Turn over 10 Co ffee, a major Brazilian export, is subject to year-t o-year price swings driv en by global demand and speculative activity. Because planting decisions today won’t fully bear fruit for several seasons, farmers often use last year’s prices as their signal for how much to plant in the coming season. ๠e price of coffee per kg in the nth year is denoted by nP . Since farmers decide how much to plant based on the previous year’s price, the quantity of coffee (in kg) produced in the nth year, denoted nS , follows 1nnSa b P  , where a and b are positive real constants. On the other side of th e market, consumers in the nth year have a demand of nnDc d P , where c and d are positive real constants. (a) When the quantity of coffee produced equals to the demand for coffee, write down a recurrence relation for nP and 1nP in terms of a, b, c and d. Hence find nP in terms of  , , n and 0P , where ca d  and b d  . [4] (b) Explain what will happen to the price of coffee in the long run if 01  . [2] For the rest of the question, use 10 6 5PP , 02P  and 1  . It is now hypothesised that farmer s base their planting decisions on the average price over the previous two years instead. (c) By rewriting the equation for nS ,