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 sequ