2025 RI Prelim+P2+Qns

@ RI 2025 [Turn Over RAFFLES INSTITUTION 2025 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 2025 9758/02/PRELIM Section A: Pure Mathematics [40 marks] 1 (a) Without using a calculator, solve the inequality 2 2 5 6 2 3 .4 2 x x x x x     [4] (b) (i) Sketch on the same diagram the graphs of lny x and 5,y x  giving the equations of any asymptotes and the x-coordinates of the points of intersection between the two graphs. [2] (ii) Hence solve the inequality ln 5.x x  [2] 2 (a) In a triangle ,ABC 2,AB angle CAB x radians and angle π 6CBA radians. (i) Show that 3 2 .ic nos sx xAC  [2] (ii) Given that x is a sufficiently small angle, show that 2,AC a bx cx   where ,a b and c are constants to be determined. [3] (b) It is given that 2 ln ln 3.xy y  Show that 22 2 2 2 d d d(2 ) 4 0.d d d y y yxy y yx x x         Hence find the Maclaurin series for ,y up to and including the term in 2.x [5]

3 @ RI 2025 9758/02/PRELIM [Turn Over 3 The terms of the sequence U are given by 1u k and 1 8 14 , 1.1 n n n uu n u   (a) For the following values of ,k describe the behaviour of the sequence U. (i) 3k [1] (ii) 10k [1] (b) Find the possible value(s) of k if the sequence U is a constant sequence. [2] The thn term of the sequence V is given by ,(1 ) 1 n n n a b av b a a n     where a and b are non-zero real constants and 1.a (c) For some values of ,a nv L as .n Find, with justification, the range of values of a for L to exist, and state the value of L in terms of a and .b [3] The nth term of the sequence W is given by when is even, when is odd. n n n u nw v n  It is given that the sequence W converges when the sequences U and V converge to the same limit. The sequence W diverges otherwise. (d) For 10,k by using part (a)(ii) and part (c), find the range of values of b for the sequence W to converge. Hence explain whether 1 r r w    is a convergent series. [3]

4 @ RI 2025 9758/02/PRELIM 4 In a computer game, a slope can be modelled as a plane p containing three points,  1, 0, 3 ,A  1, 4, 15B and  2, 3, 14 .C (a) Show that p has equation 23 1.1     r. [2] In Stage 1 of the game, Griffles travels on the slope from A along a path with equation 1 10 ,3mn                r where and m n are constants and  is a real parameter. (b) Find an equation relating and .m n [1] In the rest of the question, it is given that 1, 33m n  and Griffles is modelled as a point travelling on the slope. An obstacle in the form of a rectangular virtual screen stands on the slope with its base modelled by the line segment BC (see diagram). (c) By considering the line segment BC, show that Griffles is able to successfully navigate this obstacle without colliding into it. [3] A laser gun with a sensor is mounted at the point 5, 2, 1E above the slope. The sensor is activated when an object is within a range of 5 units. (d) Show that Griffles does not activate the sensor. [3] After completing Stage 1, Griffles is teleported to another slope. This slope can be modelled as a plane q parallel to p such that E is equidistant from and .p q (e) Find the cartesian equation of .q [3] B Virtual screen slope C A

5 @ RI 2025 9758/02/PRELIM [Turn Over Section B: Probability and Statistics [60 marks] 5 A discrete random variable X has the following probability distribution: 2P( )X x k x x   for 1, 2, 3, 4, 5,x where k is a constant. (a) Show that 1.70k [1] (b) Find the exact values of EX and Var .X [3] (c) Two independent observations 1X and 2X are taken of .X Find the probability that the difference between the two observations is at least 3. [3] 6 Kitty has 5 small, 3 medium and 2 large spherical charms and each of the 10 charms is uniquely designed. (a) Kitty arranges all the charms in a circle on a corkboard with charms of the same sizes next to each other. How many ways are there for her to do so? [2] (b) On another occasion, Kitty arranges all the charms in a line at the base of a photo frame with none of the small charms next to each other. How many ways are there for her to do so? [2] The diameters of all the small, medium and large charms are 0.5 cm, 1 cm and 2 cm respectively. Each of the spherical charms has a hole through its centre that allows it to be threaded through a chain. (c) Kitty makes a keychain that includes a 6 cm chain with one end attached to a keyring, as shown in Fig. 1. It is given that the 6 cm chain is fully threaded with charms, with no gaps between them, and is stretched taut in a straight line. For example, Fig. 2 shows the 6 cm chain threaded with 2 large and 2 medium charms. How many different ways can she make a keychain with at least one charm of each of the three sizes? [4] 6 cm chain where the charms are threaded Key ring Fig. 1 Fig. 2 Key ring 2 large 2 medium 6 cm

6 @ RI 2025 9758/02/PRELIM 7 In this question you should state the parameters of any distributions you use. A snack company produces two types of potato chips, Rays and Luffles. The mass, in grams, of a regular packet of Rays follows the distribution 2N 100,  and the mass, in grams, of a regular packet of Luffles follows the distribution N 120, 16 . The masses of all regular packets of potato chips are independent of one another. (a) If the probability of the mass of a randomly chosen packet of Rays not exceeding 2100 grams is less than 0.2, find the possible range of values of . [2] It is given that 3 for the rest of this question. (b) Find the prob