TJC 2026 Prelim P2 QP

1 @TJC 2026 9758/02 TEMASEK JUNIOR COLLEGE 2026 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 9758/02 Paper 2 14 September 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 8 printed pages and 0 blank pages. [Turn over

2 @TJC 2026 9758/02 Section A: Pure Mathematics [40 marks] 1 A sequence 1 2 3, , , ...x x x is defined by 1xa= and 1 3 2 n nnxx+ =+  , where a is a constant. (a) Find 2x , 3x and 4x in terms of a. [2] (b) Find nx in the form 1 3 2 n pq − +  , where p and q are constants in terms of a, where appropriate. [2] (c) Determine if the sequence  nx converges. [1] 2 With reference to the origin O, the points A and B have position vectors 1 7 2    and 4 2 2  − respectively. The point C divides the line segment AB such that AC : CB = 2 : 1. (a) Find the position vector of C. [1] The point D is the mid-point of OB and the point E has coordinates (1, k, 0). (b) Find the value of k such that C, D and E are collinear. [2] (c) Find the possible values of k such that the length of projection of OE  onto AB  is 10 . [3] 3 Do not use a calculator in answering this question. The complex number z satisfies the equation ( ) 2 2 i + 4i 0,z z q− + + = where q is a real number. It is given that one root is of the form p + pi, where p is a non-zero real number. (a) Find the values of p and q. [4] (b) Find the other root of the equation. [3]

3 @TJC 2026 9758/02 4 A curve C has parametric equations 2 1xt=− , 32 31y t t= − + , for t > 0. (a) Find d d y x in terms of t, and hence find the coordinates of the stationary point of C. [4] (b) Sketch the graph of C for 0t  . You do not need to find the intersection s of the curve with the axes. [2] (c) Find the equation of the tangent to curve C at the point P where 4t = . [2] (d) Using a non-graphical method, determine if the tangent to curve C at point P will meet C again. [3] 5 (a) Use the substitution 2 2sinx =+ to find the exact value of ( ) 2 2 0 4 2 d .xx−− [6] (b) The diagram below shows the two circles with equations 22 4xy+= and ( ) ( ) 22 2 2 4.xy− + − = The shaded region R is bounded by the two curves. Find the exact volume of the solid formed when R is rotated through 2π radians about the x-axis. [5] [The volume of a sphere with radius r is given by 34 .3 r ] y x R [Turn over

4 @TJC 2026 9758/02 Section B: Probability and Statistics [60 marks] 6 Anand plays a game at a funfair. In each game, he is given 3 tokens which he tosses one at a time at a target area filled with red cups and black cups. It may be assumed that the probability of him tossing a token into a red cup is a constant 0.7. Points are given for each game based on the number of tokens that are tossed into red cups. The points system is shown in the table below. Number of tokens in red cups Score 0 0 points 1 or 2 1 point 3 3 points The number of points that Anand scores in one game is denoted by S. (a) Show that ( )E 1.659S = . [3] (b) Find ( )Var S . [1] Anand wants to win a stuff toy that needs 100 points for redemption. He plays n games and is allowed to accumulate his points over multiple games. (c) State the least number of games he has to play to have a chance of winning the stuff toy. [1] (d) Find the smallest integer n for which there is at least 90% chance that Anand can redeem that stuff toy after playing n games. [4] 7 Beng, Charlie and 6 other friends bought tickets to their favourite band’s concert. These tickets are for 8 consecutive seats in a row. (a) Find the number of ways the 8 friends can seat themselves if Beng is seated next to Charlie on Charlie’s left. [1] To support their band, everyone agrees to dress in either blue or silver outfits. (b) Find the number of ways the 8 friends can choose the colour of their outfits if there must be at least two outfits of each colour. [3] After the concert, the 8 friends go to a café for dinner. The café has only 2 available tables: a round table with 5 seats and a round table with 3 seats. (c) Find the number of ways they can seat themselves if Beng and Charlie are seated at the same table. [3]

5 @TJC 2026 9758/02 8 In this question, you should state the parameters of any distributions you use. In a game, a contestant must complete two tasks A and B in the shortest time possible. The times taken to complete task A have the distribution N(m, 1.25) and the times taken to complete task B have the distribution N(7, 1). The unit for time is minutes, and the times taken to complete the tasks are independent. It is given that 3.1% of the contestants will take at least 22.8 minutes to complete both tasks. (a) Show that m = 13.0, correct to 1 decimal place. [3] (b) To win a prize, a contestant must take no longer than k mins to complete both tasks. The game master only wants to give a prize to the fastest 10% of the contestants who complete both tasks. Find the value of k. [2] (c) Find the probability that the total time taken by 3 randomly chosen contestants to complete both tasks is less than 62 minutes. [3] (d) Find the probability that the time taken by a randomly chosen contestant to complete task A is greater than twice the time taken by another contestant to complete task B. [3] [Turn over

6 @TJC 2026 9758/02 9 A production manager compiles data on rice crop yield, y tons/ha, at different amount s of rainfall, x cm. The data is tabulated as shown below. x 60 80 100 120 140 160 180 200 y 2.1 2.8 3.5 4.2 4.6 4.9 5.0 4.8 The value of the product moment correlation coefficient is 0.93244 correct to 5 significant figures. (a) What does the correlation coefficient suggest about the relationship between rice crop yield and amount of rainfall? Justify your answer. [1] (b) Sketch a scatter diagram of the data and explain why the answer in part (a) should be changed. [2] (c) Suggest a reason , in context , why a linear model y a bx=+ , where a and b are constants, is not an appropriate model for the relationship between x and y. [1] The production manager thinks that the data should be modelled by the quadratic equation ( ) 2 y c d x H= + − , where H is a suitable constant. The product moment correlation coefficient between ( ) 2 xH− and y is denoted by r. The following table gives the values of r for some possible values of H. H 182 183 184 185 r −0.997383 −0.997390 −0.997293 (d) Calculate the value of