SAJC 2026 Prelim P2 QP

ST ANDREW’S JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 MATHEMATICS 9758/02 Paper 2 14 September 2026 (Monday) Preliminary Examination 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) ______________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Answer all questions. Total marks: 100 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 7 printed pages and 1 blank page.

2 [Turn Over Section A: Pure Mathematics [40 marks] 1 It is given that 1 1 sin 2 y x = − , where 0 4x  . (a) Show that 3d cos 2dy yxx = , and hence show that 2 42d 2dy yyx =− . [3] (b) By differentiation of the second result in part (a), find the Maclaurin series for y, up to and including the term in 3x . [3] (c) Using your result in part (b) and the standard results given in the List of Formulae , deduce the Maclaurin series for 2e 1 sin 2 x x + − , up to and including the term in 2x . [2] (d) Hence find an approximate value of 0.1 2 0 e d 1 sin 2 x x x + − to 4 decimal places. [1] 2 Jane’s total salary for her first year of work at a company was $k. From the second year onwards, she is offered two salary increment schemes. Scheme A : A fixed annual increment of 35% of her total salary in the first year. Scheme B : An annual increment of r % of her previous year’s salary. (a) If Jane chooses Scheme A, show that her total salary in the first n years is ( )7 33$ 40nk n + . [2] (b) If Jane chooses Scheme B, her salary in the 5th year is $279 841. Given that the total salary she earns from the 2nd to 5th year inclusive is $918 781, find the value of r. [4] (c) Hence find the least number of years Jane is required to work in the company such that the total salary she earns under Scheme B is more than the total salary she earns under Scheme A. [3]

3 [Turn Over 3 The curve C has equation 2 by ax x=+ − , where a and b are real constants and 2x . (a) Find a condition on a and b for C to have two stationary points. [3] It is now given that 0ba . (b) Explain why C does not intersect the x -axis. [2] (c) Sketch C , stating the equations of any asymptote(s) and the coordinates of the point(s) where C crosses the axes in terms of a and b (where applicable). [3] The two asymptotes of C intersect at the point P . (d) Find the coordinates of P and show that P lies on the line L with equation 22y mx a m= + − for all real values of m . Hence state the range of values of m , in terms of a , for which L does not intersect C . [3] 4 A curve C has parametric equations 3cosx a t= , 2siny a t= , for 0 t  where a is a positive constant. (a) Sketch C. [2] (b) Find d d y x in terms of t. [1] The tangent to the curve at a point P has the equation 5 2 2 63y a x=− . (c) Find the exact coordinates of P in terms of a. Hence find the equation of the normal at P. [4] The tangent at P meets the x-axis at point Q and the normal to the curve at P meets the x axis at point R. (d) Find the exact area of triangle PQR in terms of a. [4]

4 [Turn Over Section B: Statistics [60 marks] 5 The events A, B and C are such that P( C) = 1 5 , P(A  B) = 1 10 and ( )P ' ' 'A B C = 2 5 . B and C are independent events. A and C are mutually exclusive events. Find the maximum and minimum values of P (B  C). [4] 6 In a game of musical chairs, 3 girls and 4 boys walk around a round table with six seats. When the music stops, only 6 people are seated . Find the number of different seating arrangements if (a) there are no restrictions. [1] (b) 2 particular girls are seated separately. [2] (c) boys who are seated are together. [3] 7 In an online game, players spend tokens to summon heroes. p% of the summons will result in a hero being obtained. (a) State, in context, the assumption needed for the number of heroes obtained out of n summons to be well modelled by a binomial distribution. [1] Assume now that the number of heroes obtained out of n summons has a binomial distribution. (b) The probability of obtaining exactly one hero is 6 times the probability of obtaining no heroes. Write down an equation satisfied by n and p. [1] (c) It is given that the mean number of heroes obtained is 4.8. Hence find the value of p. [2] (d) Using the value of p found in part (c), find the smallest number of summons a player must perform so that the probability of obtaining at least one hero is mo re than 0.99. [2] (e) A bundle consists of 12 summons. A player purchases 5 bundles. A bundle is classified as ‘Elite’ if it contains more than 3 heroes. Given that at least one of the bundles purchased is Elite, find the probability that exactly two bundles are Elite. [3]

5 [Turn Over 8 A discrete random variable X takes values 1, 2 and 3 with probabilities p, q and r respectively. Two independent observations of X are taken and the random variable M is defined as the maximum of the two outcomes, such that M = max (X1, X2 ). (a) Show that P(M = 2) = q(2p + q). [2] (b) Given that P(M = 1) = 0.09 and P( M = 2) = 0.55, find the exact values of p, q and r. [2] (c) Hence find the values of E(X) and Var(X). [2] (d) A random sample of 100 independent observations of X is taken. Find the probability that the sum of the 100 observations is greater than 197. [2] 9 In this question you should state clearly the values of the parameters of any normal distributions that you use. A fruit wholesaler receives two types of durians, D13 and MSW, at different times of the day. The masses, in grams, of D13 and MSW have independent normal distributions. The mean and standard deviations of these distributions are shown in the following table. Mean Standard deviation D13 250 10 MSW 550 20 On a particular day, the manager of the fruit wholesaler collected a sample of 50 durians of each type from the earliest delivery available. (a) Explain why the sample is non-random. [1] (b) Find the probability that the mass of a randomly chosen D13 durian is within 15 grams of the mean mass of D13 durians. [2] (c) It is given that the probability of the mean mass of 5 randomly chosen D13 durians and 3 randomly chosen MSW durians is more than k grams is 0.9. Find the value of k, correct to 1 decimal place. [4] The process of removing the husk reduces the mass of each MSW durian by 60%. The dehusked durians are then packed into boxes, with each box containing 4 durians. Assume that the boxes are of negligible mass. (d) Find the probability that the mass of a randomly chosen box of 4 dehusked MSW durians is at least 850 grams. [3] (e) A customer buys 6 such boxes. Find the probability that exactly 2 boxes have masses of at most 850 grams each, and exactly one box has mass of at least 880 grams. [2]

6 [Turn Over 10 A student investigates the relationship between the number of hours used of a ful