NYJC 2026 Prelim P2 QP

This document consists of 8 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2026 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/02 Paper 2 17 September 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST 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 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 9758/02 Section A: Pure Mathematics [40 marks] 1 A sequence is such that 1ua= , where a is a constant, 12a≠ , and 1 1nnuu+ = − , for n > 0. (a) Find 2u and 3u in terms of a. [2] (b) Describe how the sequence behaves. [1] (c) Find the value of a for which 101 1 49n n u = =∑ . [2] 2 The complex number iab+ is denoted by z, where a and b are real numbers, a > 0 and b < 0. (a) Find the complex number w in terms of a and b such that 2*3izw z= . [3] (b) The complex numbers z and w are represented by points P and Q respectively on an Argand diagram. Plot the points P and Q on an Argand diagram, showing clearly any geometrical relationship between them. [3] 3 A paper cup is modelled as a right circular cone of negligible thickness, with base radius r cm and height h cm. The volume of the cup is kπ cm3, where k is a positive constant. (a) Show that the curved surface area S of the cup is given by 2 2 24 2 9 .kSr r π = +  [2] [It is given that the volume of a cone of base radius r and height h is 213 rhπ and its curved surface area is ,rlπ where l is the slant height.] (b) Using calculus, find r in terms of k that gives the stationary value of S. Determine whether this stationary value of S is a minimum or maximum. [5] (c) Given that 37,k = sketch the graph showing the curved surface area of the cup as the radius varies. Find the stationary value of .S [2]

3 NYJC 2026 JC2 Preliminary Examination 9758/02 [Turn Over 4 (a) The graph of ( )fyx= is shown in Fig. 1. It intersects the x-axis at the point ( ),0a , the y-axis at the point ( )0,b , and passes through the point ( )1,1 . It is given that 1. ab<< (i) Sketch the graph of ( )1fyx −= on Fig. 1.1 in the printed answer booklet, labelling the points where the graph intersects or touches the axes. [2] (ii) Sketch the graph of ( ) 1 fy x= on Fig. 1.2 in the printed answer booklet, labelling the points where the graph intersects or touches the axes, and the equations of any asymptotes. [2] (b) A curve has equation y = g(x), where cos for 0 , g( ) 2 3 for 2 , xx x xx π πππ ≤≤= − <≤ and that g( ) g( 2 )xx π= + for all real values of x. (i) Find the exact value of 7g g(5 )2π π + . [1] (ii) Sketch the graph of g( )yx= for 03 x π≤≤ on Fig. 2.1 in the printed answer booklet, labelling the points where the graph intersects or touches the axes. [3] (iii) Sketch the graph of ( )gyx= for 03 x π≤≤ on Fig. 2.2 i n the printed answer booklet, labelling the points where the graph intersects or touches the axes. [2] x O Fig. 1 y

4 NYJC 2026 JC2 Preliminary Examination 9758/02 5 The function f is defined by 2f: 6 7xx x −+ , ,3xx∈> . (a) Explain how you know 1f− exists. [1] (b) Find 1f () x− . [2] The function g is defined by 27g: 3xx x ++ , ,x xa∈≠ . (c) State the value of a and explain why this value has to be excluded from the domain of g. [2] (d) Describe a pair of transformations that would transform the graph of ( )gyx= onto the graph of 1y x= . [2] (e) Give a reason why the composite function gf exists. [1] (f) Find ( )gf 4 . [2]

5 NYJC 2026 JC2 Preliminary Examination 9758/02 [Turn Over Section B: Probability and Statistics [60 marks] 6 An organiser of a company event is planning to present limited-edition art prints as tokens of appreciation to the guest speakers at the end of the event. Every art print is uniquely drawn by hand under three categories: Cityscape prints, Landscape prints and Seascape prints. The organiser intends to choose 6 art prints from 8 Cityscape prints, 6 Landscape prints and 5 Seascape prints. (a) Find the number of ways in which the 6 art prints can be selected so that there is at least one art print from each category and there are more Cityscape prints than from any other categories. [3] In the lucky draw segment following the presentation of tokens of appreciation, the organiser plans to prepare 9 gift boxes of distinct sizes, with 3 red, 3 blue and 3 yellow. The gift boxes are placed in a row on a table. (b) Find the number of ways in which the gift boxes can be arranged so that the row must start and end with a red box, no two red boxes are next to each other and the 3 blue boxes are all together. [3] The organiser now decides to arrange the gift boxes at random in a circle. (c) Find the probability that no 2 yellow boxes are next to each other. [2] 7 Aden and Ben each has a bag containing r red discs and 4 white discs, where 2r≥ . (a) Aden takes 3 discs at random from his bag, one at a time and without replacement. The events A and B are defined as follows. A: the first disc drawn is white B: the total number of white discs drawn is exactly 1 (i) Find P( )AB∩ in terms of r. [1] (ii) Given that A and B are independent events, show that 8.r = [2] For the rest of the question, let 8.r = (b) Ben takes discs at random from his bag, one at a time and without replacement, until a red disc is obtained. The random variable X is the total number of draws required to get the first red disc. Determine the probability distribution of X, giving the probabilities in exact form. [3] All the discs removed earlier are returned to their respective bags. Aden and Ben each takes discs at random from his own bag, one at a time and without replacement, until a red disc is obtained. (c) Given that there is exactly one person who takes at most 2 draws to get the first red disc, find the probability that Ben gets his first red disc on his third draw. [3]

6 NYJC 2026 JC2 Preliminary Examination 9758/02 8 At a cafe, each customer receives one reward coupon. The coupon may yield a monetary reward in the form of points according to the following distribution. Reward None Silver Gold Points 0 1 4 Probability 0.4 p 0.6 p− Let Y be the number of points earned per coupon. (a) It is given that the expected number of points earned from one coupon is 1.8. Find p. [1] A coupon is called redeemable if it yields at least 1 point. Ten customers are selected independently. Let X be the number of redeemable coupons among the ten customers. (b) Given that at least 3 customers have redeemable coupons, find the probability that at most 7 customers have redeemable coupons. [3] The cafe operates for 80 days. L et iX denote the number of redeemable coupons among ten randomly selected customers on the ith day. Let 80 1 i i TX = =∑ . (c) State the mean and variance of T. [2] Let W be the average number of redeemable coupons per customer per day. (d) Explain why W is approximately normally distributed. [1