NYJC, TJC, VJC 2026 Prelim P2

This document consists of 6 printed pages and 2 blank pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2026 [Turn Over NANYANG JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 18 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 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.

2 NYJC 2026 JC2 Preliminary Examination 9649/02 Section A: Pure Mathematics [50 marks] 1 The linear transformation 33:T →  is defined by : xx T y y zz                  M  , 3 1 1 where 1 3 1 1 1 3 − = − − −M . (a) Find the dimension of the null space of T. [2] (b) It is given that 1 2 3 1 0 1 1 1 0 01 ,, 1                       === −     v v v . Determine with a reason, whether 23( ), ( ), ( )T T T1v v v form a basis for 3.  [2] (c) For matrix M , state the eigenvalue corresponding to the eigenvector , 1 0 1   −=u and determine an eigenvector corresponding to another eigenvalue. [3] 2 Anand and Beng set up fish farms on two identical, adjacent plots of land such that the fish populations in both farms may be modelled by the same differential equation d 21d 25PP PHt = − − , where P is the population of fish (in thousands) and H is a constant (in thousands) representing the change in fish population due to harvesting. One unit of t represents one harvest season. Anand started his farm with 4000 fish, but the fish population in his farm died out after several years. Beng started his farm with 16500 fish, and the fish population in his farm grew and stabili sed at 20000 after many years. (a) Find the number of fish harvested each season. [2] (b) On the same graph, illustrate how the populations of fish in Anand’s and Beng’s farm vary over time. [2] (c) Find the smallest number of fish Beng could have started his farm with to reach the same population size after many years. [2]

3 NYJC 2026 JC2 Preliminary Examination 9649/02 [Turn Over 3 Amy wishes to find the root of the equation 5 1.0093 0xx− − = (*) First, she tries using a Newton-Raphson iteration method with 0 0x = . (a) Find the values of 1 2 3,,x x x and 4x , and explain whether A my will be able to find a solution for the equation. [3] (b) Suggest what A my should have done before choosing an appropriate initial value for h er iterative process. [1] (c) Without using a graphing calculator, sketch two suitable curves to show that (*) has exactly one real positive root, justifying your answer. [1] Next, Amy decides to find the root of (*) using a fixed-point iteration formula 1 4 1.0093n n n xx x + += with 0 1x = . (d) Sketch a graph showing how 1x and 2x are obtained using this method, and explain why Amy will not be able to find a solution to this equation. [2] (e) Rewrite the equation (*) in a suitable form and hence use the iterative formula ( )1 gnnxx+ = to find the solution of (*) correct to 3 significant figures using 0 1x = . [3] 4 The polar equations of two curves are given by 1 : 1 sinCr =+ and 2 : 1 cos2Cr =+ for ππ −   . (a) Find the exact polar coordinates of all point(s) of intersection of 1C and 2C . [3] (b) Find the exact area of the region of 1C that lies inside 2C . [4] Let P be the point of intersection of 1C and 2C in the first quadrant. A straight line parallel to the initial line passes through P and intersects 2C at another point, Q, in the first quadrant. (c) Show that at Q, 13 1sin 4 −= . [4] (d) The region bounded by the line segments OP, OQ and the arc of the curve 2C between P and Q forms a polar sector. Find the total perimeter of this sector, giving your answer correct to 4 decimal places. [2]

4 NYJC 2026 JC2 Preliminary Examination 9649/02 5 A mathematical model is given by the differential equation 2 d e 2 d1 yyx xx − −= + where 0y= when 1x= . (a) Using a step size of 0.5, obtain an approximation for y when 2x= using (i) Euler’s method, [2] (ii) the Improved Euler’s method. [3] Show all your steps clearly. (b) It is later found that the actual value of y is −0.5108256, correct to 7 significant figures, when 2x= . Calculate the percentage error of the estimates obtained by both methods. [2] (c) Give a reason why the Improved Euler’s method estimate gives a larger error than the Euler’s method estimate in this case. [1] (d) Suggest a scenario when the Euler method and Improved Euler method may give estimates that are very close to each other. [1] (e) Using the substitution lnyu= , find y in terms of x and sketch the solution curve for 03 x . [5] Section B: Probability and Statistics [50 marks] 6 Under usual operating conditions, a help desk receives on average one call every 2.5 minutes. It may be assumed that calls are received at random and that the probability of two or more calls being received simultaneously is negligible. (a) State the two further assumptions needed for the number of calls received in a fixed period of time to be modelled by a Poisson distribution. [2] Assume now that a Poisson model is valid, and that usual operating conditions apply. (b) Find the probability that no call is received in a period of 5 minutes. [1] Let T minutes denote the waiting time until the next call is received. (c) State fully the distribution of T. [1] (d) Find the minimum value of t, correct to 1 decimal place, in order to be 99% certain that the next call is received within t minutes. [2] (e) A help desk operator has already waited 5 minutes without receiving a call. Find the expected additional waiting time until the next call is received. [2]

5 NYJC 2026 JC2 Preliminary Examination 9649/02 [Turn Over 7 Benford’s Law states that, in many tables containing large numbers of numerical values, the probability distribution of the leading non-zero digit D is given by 10 1P( ) log , 1, 2, ... , 9.dD d d d+= = =  Fabricated financial data often deviate from Benford's Law and auditors use it as an initial analytical procedure to identify datasets that may require closer examination. An auditor takes a random sample of 200 invoice amounts from a branch that is being reviewed. The data are as follows. Leading digit 1 2 3 4 5 6 Observed frequency 52 34 22 16 38 38 (a) Show that the theoretical probability for the pooled category 6D is 10 5log 0.221853  . [1] (b) Carry out an appropriate test to determine if there is very strong evidence that the leading non-zero digit distribution does not follow Benford’s Law. [5] (c) Calculate the percentage of the total test statistic represented by the contribution of the cell corresponding to 5D= . Discuss why the result of the test in part (b) may not be sufficient to conclude that the invoice amounts from the branch is f