ACJC, NJC, SAJC 2026 Prelim P2

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 17 September 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all the 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 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. __________________________________________________________________________________ This document consists of 10 printed pages. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/02 Section A: Pure Mathematics [50 marks] 1 For real k, let 3 : 0 and 2 3 0 and 2 3 0k x S y x y z x y z x y kz z  =  + + = + + = + + =  . (a) Determine the value of k for which kS contains non-zero vectors. [2] Let k take the value found in part (a). (b) Solve the equations defining kS and give kS in vector form. [1] (c) Hence solve the system of equations 1, 2 3 1, 2 3 2. x y z x y z x y kz + + = + + = + + = Determine whether the solution set T is a subspace of 3  . [2] 2 (a) Solve the differential equation ( )d 11dyx y xx+ − = , expressing y in terms of x. [3] (b) Sketch, on separate diagrams, a member of the family of solution curves for each of the following conditions and state the equation of each curve. (You are not required to label the coordinates of any stationary points.) (i) A solution curve with a vertical asymptote and no stationary point; (ii) A solution curve with a vertical asymptote and at least one stationary point. [3] (c) State the equation of the solution curve that does not have any vertical asymptote, and explain why this curve does not have any vertical asymptote. [2]

3 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 3 The temperature T at any point ( ),xy is given by ( ) 22, 4 5T x y x xy y x y= + + − − , for all real values of x and y . A particle is at the point ( )1,1P . (a) Find a direction in which the temperature increases most rapidly at P. Hence, find the maximum rate of increase of temperature at P. [3] A unit vector u in the xy -plane makes an angle  with the positive x -axis. (b) Express u in terms of  . Given that the directional derivative of T at P in the direction of u is zero, find the value of  for 0 π , giving your answer exactly. [3] The surface S has equation ( ),z T x y= . (c) Find the Cartesian equation of the tangent plane to S at the point ( )1,1, 6− . [2] (d) Find the vector equation of the normal line to S at the point ( )1,1, 6− . [2]

4 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/02 4 It is known that the function ( ) 52f 3 8 4x x x= − + has exactly one negative root,  , and exactly two positive roots ,  and  , such that  . (a) Show that 0.75 1 1.25    . [2] (b) Use Newton-Raphson iteration with an initial value 1− to estimate  to 3 decimal places. [2] A student, Annie, wishes to explore different iterative approaches to estimating  and  using initial values close to 1. (c) Explain why Newton-Raphson iteration may not be appropriate for estimating either  or  with an initial value is close to 1. [1] Her classmate, Bhaven, proposes using the iterative formula 4 1 31 82 nn n xx x+ =+ with a positive initial term 1x , to estimate  and  . (d) Show that if the sequence  nx converges to a limit using Bhaven’s formula, then it must converge to either  or  . [2] (e) Show that Bhaven’s formula is suitable for estimating  , but not  . Hence, estimate  to 3 decimal places. [4] (f) Suggest an iterative method to estimate  with an initial value of 1 , and carry out the first step of the suggested method. [2]

5 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 5 Researchers wish to model the amount of atmospheric carbon dioxide generated due to human activities, known as anthropogenic carbon. In recent years, technological advancements have been made to store anthropogenic carbon and mitigate climate change. These researchers have proposed the general model 1 1 100n n nC C S + = + − for 0n , where •  is a positive constant denoting the percentage increase in anthropogenic carbon generated per year; • nC denotes the amount of anthropogenic carbon present at the start of the nth year; • nS denotes the amount of anthropogenic carbon stored during the nth year, and • 0C denotes the amount of anthropogenic carbon present initially. In a constant storage model, it is taken that nSk= , where k is a positive constant. (a) (i) Express nC in terms of  , k and 0C . [3] (ii) State a condition on  , in terms of k and 0C , to prevent the long run amount of anthropogenic carbon from increasing without bound. [1] Now, assume an idealised technological model, where it is taken that • For 0n= , 1 00 200SC = , • For 1n , nS is half of the sum of the excess anthropogenic carbon generated in the nth year and the year before. (b) (i) Find 1C and show that 11 10 200 200n n nC C C +− − + + = for 1n . [2] (ii) Find an expression for nC , in terms of 0C and  , and hence determine the exceptional value of  for which the model predicts a linear increase in anthropogenic carbon. Deduce exactly the range of values of  such that the long run amount of anthropogenic carbon does not exceed 01.5C . [6] (iii) Give a criticism of the idealised technological model and suggest an improvement to the model to overcome the issue. [2]

6 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/02 Section B: Statistics [50 marks] 6 A study was conducted to investigate whether a dietary intervention programme or a standard exercise programme is more effective in reducing stress levels among university students in Singapore. Eighteen students with elevated stress scores were divided into nine pairs with similar baseline scores. For each pair, one member was randomly assigned to the dietary intervention programme and the other to the standard exercise programme. Eight weeks later, the following stress scores were recorded, all lower than their baseline scores. (Scores are measured on a standardised scale, where higher scores indicate greater stress.) Pair 1 2 3 4 5 6 7 8 9 Dietary intervention 38 50 47 54 44 55 49 36 41 Standard exercise 41 46 52 61 53 55 47 42 51 Carry out the following tests at the 5% significance level to test whether the dietary intervention programme is more effective in reducing anxiety than the standard exercise programme. (a) Sign test; [5] (b) Wilcoxon matched-pair signed rank test. [3] Comment on the use of the above tests. [1]

7 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 7 A streaming platform conducted a survey to study the genre preference of its subscriber