2025 JPJC H2 Math Prelim P2 QP

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2025 MATHEMATICS 9758/02 Higher 2 16 Sept 2025 Paper 2 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 7 printed pages and 1 blank page. [Turn over Answer all 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 co rrect 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 by [ ] at the end of each question or part question .

2 Section A: Pure Mathematics [40 Marks] 1 To prepare for the General Election, a political party studies the effectiveness of spreading information on Instantglam , a social media platform , by uploading a particular video. The number of Instantglam users who have viewed the video increases at a rate that is directly proportional to the number of Instantglam users who have yet to view the video. The number of Instantglam users who have viewed the video t hours after the video is uploaded is x. The total number of Instantglam users is P, assumed to be fixed. (i) Write down a differential equation and show that ( )1e ktxP −=− , where k is a positive constant. [4] It is known that 12 hours after the video is uploaded, half the total number of Instantglam users have viewed the video. (ii) Estimate to the nearest hour, the time needed for 80% of the total number of Instantglam users to have viewed the video. [3] (iii) Sketch the graph of x against t. [2] 2 With reference to the origin O, the points A, P and Q have position vectors a, p and q respectively. A straight line l through the point A is parallel to a unit vector e. (i) The point Q lies on l. Show that ( )−  =q a e 0 . [2] (ii) Given that P is not on l, give the geometrical meaning of ( )−p a e . [1] It is also given that 3=pq and AQ is 2 units. (iii) Using the results in (i) and (ii), or otherwise, show that the area of triangle APQ can be written as k qe , where k is a constant to be determined. [4] (iv) Hence, or otherwise, find the acute angle between l and PQ, given further that the area of triangle APQ is 3 units2 and 2=q . [3]

3 3 The curve C has parametric equations sin 1x =+ and 3 cos 1y =− , where ππ 22 −   . (i) Show that d 3 tand y x =− . [2] (ii) The tangent to C at point ( )sin 1, 3 cos 1T t t +− makes an angle of 3π 4 with the positive x-axis. Find the equation of the normal to C at T. [3] (iii) Find the exact y-coordinate of the point on C where the tangent to C is parallel to the x-axis. [2] (iv) Find the area of the quadrilateral bounded by the axes, the normal to C at T and the tangent in (iii). [3] 4 Do not use a calculator in answering this question. (a) One of the roots of the equation 325 68 0w pw w q+ + + = , where p and q are real, is 3i− . Find the other roots of the equation and the values of p and q. [5] (b) Two complex numbers are given by 1 1 2iz =− + and 2 2iz =+ . Draw an Argand diagram showing 1z and 2z , labelling the origin as O and the points representing 1z and 2z as 1Z and 2Z respectively. Given that 3 1 2z z z=+ , mark the corresponding point 3Z on your Argand diagram, showing clearly the geometrical relationship between 1Z , 2Z and 3Z . [2] (i) Find 1 2 z z in the form ik . [2] (ii) Hence, or otherwise, show that 2 3 1OZ Z Z is a square. [2] [Turn over

4 Section B: Probability and Statistics [60 Marks] 5 In a strategy game called Colour Clash, players draw coloured orbs from a magical pouch to determine their elemental powers. The pouch contains: • 5 red orbs (Level 1 to 5), • 4 blue orbs (Level 1 to 4), • 3 green orbs (Level 1 to 3). Orbs of the same colour are non-identical with each same-coloured orb representing a different level. A player randomly draws four orbs without replacement at the start of the game. Given that the order in which the orb is drawn is not relevant, find the number of ways this can be done such that (i) there are no restrictions, [1] (ii) there are at least two colours present, [2] (iii) at least one orb of each colour is drawn. [3] 6 A particular streaming service platform claims that the average time a user spends watching content on the platform per visit is more than 15 minutes. A team of data analysts for the platform wants to verify this claim. They randomly select a sample of 80 users and record the amount of time each user spends watching content in a single visit. The sample mean of time spent is found to be 16 minutes. The population standard deviation of time spent is assumed to be  minutes. (i) Explain why the team should carry out a 1-tail test. [1] A test at the 5% significance level indicates that the platform’s claim is valid. (ii) Explain why the team is able to carry out a hypothesis test without knowing anything about the distribution of the times spent by the users. [1] (iii) Find the range of values of  . [4] (iv) Explain, in the context of the question, the meaning of 'at the 5% significance level'. [1]

5 7 In a game, a player selects one ball at random from a bag which contains four balls numbered ‘1’, ‘1’, ‘2’ and ‘3’. The player notes the number on the selected ball , and then tosses that number of fair coins . The number of heads obtained is denoted by X and has a probability distribution as follows: x 0 1 2 3 ( )P Xx= a 15 32 b 1 32 (i) Find the exact values of a and b. [3] (ii) Find E( )X and show that ( ) 39Var 64X = . [2] (iii) The player plays 50 games. F ind the probability that the average number of heads obtained per game exceed 1. [2] 8 A company makes portable speakers. Some speakers turn out to be faulty. Each carton has 24 portable speakers. (i) State, in context, two assumptions needed for the number of faulty portable speakers in a randomly chosen carton to be well modelled by a binomial distribution. [2] It is given that 2% of the portable speakers produced are faulty. (ii) Find the probability that a randomly chosen carton contains at most one faulty portable speaker. [1] (iii) A carton with more than one faulty portable speaker is considered as ‘substandard’. Find the probability that, out of 50 cartons,