DHS, HCI, RI, TMJC 2026 Prelim P2

© HCI 2026 HWA CHONG INSTITUTION 2026 JC2 Preliminary Examination Higher 2 9649/02 22 September 2026 3 hours Total marks: 100 Candidate Name CT Group 25 READ THESE INSTRUCTIONS FIRST Do not write anything on the List of Formulae (MF27). Answer all the questions. Write your answer to each question in the space provided in 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 must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part of question. FURTHER MATHEMATICS Question Paper This document consists of 7 printed pages and 1 blank page. Remarks for Candidates: a) INSTR: Did not follow instructions stated in the question (e.g. correct significant figures, exact values, coordinates, required form, etc.). b) NOT: Incorrect or inappropriate use of mathematical notation. c) ACC: Inaccurate answer (e.g. due to premature rounding, omission of “+C” in indefinite integrals, etc.).

2 © HCI 2026 Section A: Pure Mathematics [50 marks] 1 The sequence  na is given by 21 0n n na a a+++ + = , where 121, 1 and 2n a a = = . (a) Find na in terms of n, giving your answer in the form ( ) ( )cos sinP n Q n+ where P, Q and  are constants to be determined. [4] (b) Determine the set of all possible values that na can take as n varies. [1] (c) Deduce the value of 2026 1 n n a =  . [2] 2 (a) Consider the gradient function ( )d fdy xx = . y appl ing the trapeiium rule to approximate the integral of ( )f x over the interval ( ),1x kh k h+ , derive the improved Euler formula ( ) ( )( )1 f f 12kk hy y kh k h+  + + + where ky denotes the value of y when x kh= . [3] (b) A function y satisfies d 1 1 d y x x y=+ , where 2y= when 3x= . sse two steps of the improved Euler method to calculate an approximation of the value of y, when 3.2x= , giving our answers to three decimal places. [4] 3 (a) (i) If z is a complex number and *z is its conjugate, show that 2 *zz z= . [1] (ii) Let z and w be complex numbers which do not lie on the unit circle. Given that 2 12*zwzw− =− , find the value of z . [4] (b) The complex number iz x y=+ where x, y are real , satisfies the locus L , which is defined by 2iz k z k− = − where k is a positive real constant. (i) Show that the cartesian equation of the locus L can be expressed as 4 2 3yxk−= . [2] (ii) A circle C is defined by the equation 3 6i 20.z+ − = It is given that the point on L which is nearest to the origin lies on the circumference of circle C . Find the possible value(s) of .k [4]

3 © HCI 2026 [Turn over 4 Let 33:T →  be the linear transformation representing the anticlockwise rotation about the y axis through an angle  and let 23, and 1e e e be the standard basis vectors of 3  , where 23 1 0 0 0 , 1 and 0 0 0 1            = = =               1e e e . The diagram below illustrates the transformation s of 1e and 3e to ( )1T e and ( )3T e respectively. (a) Explain why ( ) cos 0 sin T    =−1e , ( ) sin 0 cos T    =3e and find ( )2T e . [3] 33:S →  is a linear transformation that scales a vector by non-zero scale factors of  and  along the x and z axes respectively. Let A and B be the matrices representing T and S respectively. (b) Give a geometrical interpretation of 1−A and hence find 1−A and 1−B . [3] A curve 1C in the xz-plane has an equation 222 36xz+= . Another curve 2C is obtained by rotating 1C 30 anticlockwise about the y-axis followed by scaling parallel to the x-axis and z- axis by scale factors of 2 and 3 respectively. (c) Hence find a cartesian equation of 2C , leaving your answer in a simplified form. [4] y T(e1) T(e3) P z x

4 © HCI 2026 5 (a) Solve 3d 2ed xux u xx −= , leaving our answer for u in terms of x. [3] Sketch, on a single diagram, three distinct members of the famil of solution curves. [2] (b) sse the substitution y = xz to show that ( ) 2 2 2 d 2gd zx z x xx −= can be written as ( ) 2 2 dd 2gddyyx x xxx−= . [2] (c) Solve the differential equations (i) 2 23 2 d 2ed xzx z xx −= , [4] (ii) 2 2 2 d 21d zxz x −= . [4] giving each answer for z in the form ( )fzx= .

5 © HCI 2026 [Turn over Section B: Probability and Statistics [50 marks] 6 A publisher is interested in testing whether the number of printing errors per 100 pages follows a Poisson distribution. The observed frequencies from examining 400 randomly selected sets of 100 pages are as follows: Number of errors 0 1 2 3 4 5 6 7 8 Frequency 80 100 86 67 33 25 6 2 1 (a) Test at the 5% level of significance whether the Poisson distribution provides a good fit for the number of printing errors per 100 pages. [5] (b) Referring to the two largest contributions to the test statistic, discuss what the test indicates about the printing errors. [1] 7 The probabilit densit function (p.d.f) of a normal distribution is given b ( ) 2 22 2 1f( ) e , 2π x xx    −− =  where  and 2 are the mean and variance of the normal distribution respectivel . (a) ssing the given p.d.f, show that 22 0 πed 8x x − = . [2] A continuous random variable X has its probabilit distribution given b ( ) 22e , 0,g 0 otherwise, xkxx − =  where k is a constant. (b) Sketch the graph of g( )x . [1] (c) ssing the result found in part (a), find the value of k . [1] (d) With the value of k found in part (c), find Var( )X , leaving our answer in exact form. [3] (e) Find the median value of X. [2]

6 © HCI 2026 8 A biased blue die is rolled. The probabilit that “1” is obtained is 0.4 . Let X be the number of attempts to obtain the first “1” on the blue die. (a) Find the probabilit that at least 3 attempts are required to obtain a “1”. [1] (b) Given that no “1” was obtained in the first 6 attempts, find the probabilit that the first “1” is obtained on the 10th attempt. [2] Another biased red die with the same probabilit of obtaining “1” is rolled independentl . Let Y be the number of attempts to obtain the first “1” on the red die. Let W be the minimum of X and Y . (c) Find ( )P Ww= , in terms of w . Hence or otherwise, find ( )P2W = . [4] (d) Let T X Y=+ . Find ( )P4T = . [2] (e) Using your result from parts (c) and (d), show that T and W are not independent. [2] 9 A sugar company produces bags of sugar with a target weight of 1 kg per bag. The mass of the packaging is negligible. A quality control team randomly selects 40 bags and records the mass of the sugar in each bag in grams. The data are summarized as follows. 39889.4x= 2 39779879.2x = (a) Construct a 95% confidence interval for the mean mass of t