TJC_9758_2025_Prelim_P2

1 ©TJC 2025 9758/02 TEMASEK JUNIOR COLLEGE 2025 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/02 16 Sep 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (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 7 printed pages and 1 blank page. [Turn over

2 ©TJC 2025 9758/02 Section A: Pure Mathematics [40 marks] 1 It is given that 3ln 1 e xy=+ . (a) Show that 23 d2e =3ed yx y x . [2] (b) By further differentiation of the result in part (a), find the Maclaurin series for y , up to and including the term in 2x , giving the coefficients in exact form. [4] (c) Deduce the Maclaurin series for 31eln 2 x− + , up to and including the term in 2x . [2] 2 (a) Using the result 2 2 1 1 ,(2 1)(2 3) 5 2 3 n r r r n= =−+ + + find 2 2 2 (2 1)(2 1) n r rr= −+ in terms of n. [4] (b) A sequence of positive numbers 1 2 3, , ,......x x x satisfies the recurrence relation 1 4 5 for 1,2,3, .nnx x n+ = + = (i) Given that the sequence converges to L, find the exact value of L. [2] (ii) Prove that 22 1 5( ).nnx L x L+ − = − [1] (iii) Use the result in part (ii) to show that if ,nxL then 1 .nxL+  [2] 3 The function f is defined by 1f : e 2 xx −+ , , 0xx . (a) Sketch the graph of ( )fyx= , ( ) 1fyx −= and ( ) 1ffyx −= on the same diagram, showing clearly their relationship. [3] The function g is defined by 2g : 1 4 , 2x x x x+ −  . (b) Explain why 1g− exists. [1] (c) Find 1g ( )x− and state its domain. [3] (d) Explain why the composite function gf exists. [2] (e) Find the exact range of gf. [2]

3 ©TJC 2025 9758/02 4 [The arc length L of a sector of radius a and angle  is given by L = a ; The volume V of a cone of base radius r and height h is given by V = 21 3 rh ] A metal sheet is shaped as a circular sector with angle  and fixed radius a, as shown in Figure