CJC 2026 Prelim P2 QP

9758/02/J2PRELIM/2026 [Turn Over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/02 Paper 2 16 Sep 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. This document consists of 6 printed pages.

2 9758/02/J2PRELIM/2026 Section A: Pure Mathematics [40 marks] 1 The vectors v and w are defined as 2 v a b and 5 4 w a b where a and b are unit vectors. Given that v and w are perpendicular, find the angle between a and b. [4] 2 An arithmetic progression A has first term a and non -zero common difference d. A geometric progression G has first term a and common ratio r, where 1r  . It is given that the first, fourth and thirteenth terms of A are equal to the first three consecutive terms of G respectively. (a) Express d in terms of a and show that r = 3. [4] (b) Let Sn denote the sum of the first n terms of A. Given that 0a  , find the smallest value of n such that 100nS a . [3] (c) A new sequence U is defined such that the nth term is given by n n n ku T , where nT is the nth term of the geometric progression G and k is a positive constant. Show that U is also geometric. [2] (d) Given that U is a convergent sequence, find 1 r r u    in terms of a and k. Express your answer as a single fraction. [2] 3 Given that 3 i is a root of the equation  P 0z  , such that   4 3 2P 2 14 10z z z az bz     , where a and b are real constants. (a) Find the values of a and b. [3] (b) Using the values of a and b found in (a), state another complex root of the equation in the form x + iy, where x and y are real numbers and 0.y Justify your answer. [2] (c) Find, using an algebraic method, the other roots of  P 0z  . [3] (d) Indicate clearly, on an Argand diagram, the coordinates of the points that represent the roots of  P 0z  . Hence, state the shape of the quadrilateral formed by these points, justifying your answer. [4]

3 9758/02/J2PRELIM/2026 [Turn Over 4 The points A 0, 2,1 , B 8, 6, 0 and C 4,1,1 lie on a plane 1 . (a) Find a cartesian equation for 1 . [4] The point T (1, 2, k) lies outside 1 , where k is a real constant. (b) Show that the shortest distance from T to 1 is 1 8 99 k . [2] The plane 2 has a cartesian equation given by 2 2 7x y z   . (c) Given that T also lies outside 2 and is equidistant from the planes 1 and 2 , find the possible values of k. [5] (d) Find an equation of the line of intersection between 1 and 2 . [2]

4 9758/02/J2PRELIM/2026 Section B: Probability and Statistics [60 marks] 5 Vis plays a game where he puts a $1 coin into a coin-operated machine to start the game. The possible outcomes of each game are as follows:  Vis loses the game with a probability of 0.8 and the machine pays out nothing,  Vis draws the game with a probability of 0.15 and the machine pays out a $1 coin,  Vis wins the game with a probability of 0.05 and the machine pays out ten $1 coins. The game stops after each outcome. The outcomes can be modelled by a random variable Y representing the number of $1 coins profited at the end of one game. (a) Show that  E 0.35Y  and find  Var .Y [3] Vis starts off with ten $1 coins and decides to play the game exactly ten times. At the end of ten games, find (b) the expected number of $1 coins that Vis will have, [2] (c) the probability that Vis will have at least ten $1 coins left. [3] 6 A railway company sells 200 non-reserved tickets for a train that has 180 seats. Every ticket holder who turns up gets a seat, until all seats are filled. Historical records show that on average, each ticket holder turns up for the journey with probabilit y 0.88. Let X represent the number of ticket holders who turn up. (a) State, in context, two assumptions needed for X to be well modelled by a binomial distribution. [2] (b) Explain why one of the assumptions stated in part (a) may not hold in this context. [1] Assume now that X follows a binomial distribution. (c) Find the probability that every ticket holder who turns up gets a seat. [2] (d) Given that at least one ticket holder who turns up is unable to get a seat, find the probability that exactly three ticket holders who turn up are unable to get a seat. [2] Instead of selling 200 tickets, the railway company now wants to ensure that every ticket holder who turns up gets a seat on the train more than 95% of the time. (e) Determine the largest number of tickets that the company can sell. [3]

5 9758/02/J2PRELIM/2026 [Turn Over 7 Ana and Din travel with one child each. Together with a couple consisting of one husband and one wife, they sit in a minibus with six passenger seats as shown. Find the number of ways the six of them can be seated if (a) there are no restrictions, [1] (b) Ana and Din must sit beside their respective child, [2] (c) the wife refuses to sit in the seat directly behind Ana. [2] During lunch, the six of them sit at a round table with six identical seats. (d) In how many ways can Ana sit directly opposite the wife? [2] Seven bowls of ice-cream of four flavours are served for dessert: 1 coffee, 2 chocolate, 2 mint, and 2 vanilla. Each bowl of the same flavour is indistinguishable and each person must select exactly one bowl. (e) In how many ways can the bowls of ice -cream be distributed if the two children choose the chocolate flavour? [3] 8 A researcher wishes to investigate the average time taken by students to complete a certain test . He decides to carry out a hypothesis test with a random sample of 40 students. The time, x minutes, of the random sample of 40 students are recorded and summarised as follows. 240 1080 31100n x x     (a) Calculate unbiased estimates of population mean and variance for the time taken by students to complete the test. [2] (b) Given that the test concludes there is sufficient evidence at the % significance level that the average time taken by students was not 25 minutes, find the range of possible values of . [4] (c) Using the same sample, the test concludes there is sufficient evidence at the 2.5% significance level that the average time taken by students is less than k. Find the range of possible values of k. [3] (d) Explain why this test would be inappropriate if the researcher had taken a random sample of 1