RI 2026 H2 Math Prelim P2 Qns

@ RI 2026 [Turn Over RAFFLES INSTITUTION 2026 YEAR 6 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS Paper 2 9758/02 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. This document consists of 9 printed pages and 1 blank page. RAFFLES INSTITUTION Mathematics Department

2 @ RI 2026 9758/02/PRELIM Section A: Pure Mathematics [40 marks] 1 (a) The complex number w is such that 2* 6 2 3 iw ww  . Find exactly the possible values of ,w giving your answer in the form ix y where x and y are real numbers. [3] (b) The complex numbers found in part (a) are represented by points A and D in an Argand diagram as shown below. Points B and C represent complex numbers band c respectively. Given that ABCD is a rectangle with2AD AB, find complex numbers b and ,c giving your answers in the form iu v, where u and v are real numbers. [3] (c) By finding arg( )b, find the exact value of 5tan12. [2] 2 (a) Find 22 3 d2 3xxx x . [3] (b) Using the substitution tan ,2xu show that 21 2 d d2 cos 3x ux u  . Hence find the exact value of 201 d .2 cosxx [6] C B Re O A Im D

3 @ RI 2026 9758/02/PRELIM [Turn Over 3 The function f is given by f : e , x ax a x    , where a is a positive constant. It is given that the range of f is ,k , where 0k . (a) Find the range of values of a. [2] (b) Show that 1f does not exist. [2] (c) If the domain of f is restricted to ,m , state the largest value of m in terms of a for which 1f exists. [1] (d) Using your answer found in part (c), find 1f in similar form. [3] The function g is given by g : ln 1 , .xx x a a      (e) Show that the composite function gf exists and find the range of gf in terms of a. [3]

4 @ RI 2026 9758/02/PRELIM 4 Do not use a calculator in answering this question. (a) The curve f ( )y x passes through the origin and has gradient given by d d 1 y x y x xy   . (i) Show that 22 2 d d d(1 ) ( 1) 1d d d y y yxy x yx x x         . [2] (ii) By further differentiation of the result in part (a)(i), show that 23 2 2 3 2 2 d d d d d(1 ) p( ) q( ) 0d d d d d y y y y yxy x y rx x x x x                   , where the functions p(x), q(y) and the constant r are to be determined. [3] (iii) Hence find the Maclaurin series for y up to and including the term in 3x . [2] (b) The equation 2 2 cos 3sin 2 3 04x x       has a root , which is close to zero. (i) Show that 2 0a b    , where a and b are integers to be determined. [3] (ii) Hence find an approximation for  in surd form. [2]

5 @ RI 2026 9758/02/PRELIM [Turn Over Section B: Probability and Statistics [60 marks] 5 A random variable X has probability distribution given by 2 for 1, 2,3, P( ) (7 ) for 4,5,6, 0 otherwise, kx x X x k x x      where k is a positive constant. (a) Show that 1 20k . [2] (b) Find the values of E X and VarX . [3] Another random variable Y is defined by 1 23Y X X  , where 1X and 2X are independent observations of X. (c) Find VarY . [2] 6 (a) For a special opening act in a talent show, 8 performers (3 singers, 3 dancers and 2 musicians) are to stand in a circle on stage. Find the number of possible arrangements if (i) there are no restrictions, [1] (ii) all performers of the same role must stand together. [2] (b) For the main performance, 12 performers (3 singers, 5 dancers and 4 musicians) are arranged in a line on stage such that no two musicians are adjacent. Find the number of possible arrangements. [2] (c) 7 performers are selected from 15 performers (6 singers, 5 dancers and 4 musicians) for the closing act. Find the number of possible selections if there must be at least 2 performers from each role. [3]

6 @ RI 2026 9758/02/PRELIM 7 In this question, you should state the parameters of any distributions you use. In a running competition, the time taken by members from the Sports Recreational Club to complete a 20 km race is normally distributed with a mean of 105 minutes and a standard deviation of 12 minutes. (a) Find the probability that a randomly chosen Sports Recreational Club member finishes the 20 km race in less than 100 minutes. [1] (b) Find the time that is exceeded by 10% of the Sports Recreational Club members. [1] Members from the Swift Runners Club competed in the same 20 km race. After analysing their results, it was found that 36% of them completed the race in less than 95 minutes and 8% of them completed the race in more than 125 minutes. You may assume that the Swift Runners Club members’ running times can be modelled using a normal distribution with mean  and standard deviation . (c) Determine the values of  and  . [4] (d) Using the values of the parameters found in part (c), find the probability that the total running time of 2 randomly chosen Swift Runners Club members is less than 1.8 times of the running time of a randomly chosen Sports Recreational Club member. State an assumption you have made in your calculation. [4]

7 @ RI 2026 9758/02/PRELIM [Turn Over 8 A light source is placed at varying distance