NYJC_9758_2025_prelim_P2

This document consists of 6 printed pages and 2 blank pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2025 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CT CLASS 2 4 Centre Number/ Index Number / MATHEMATICS 9758/02 Paper 2 17 September 2025 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. The total number of marks for this paper is 100.

2 NYJC 2025 JC2 Preliminary Examination 9758/02 Section A: Pure Mathematics [40 marks] 1 (a) Find the set of values of x for which 3 123 xx− − . [2] (b) Without using a calculator, solve 2 18 15 14 x xx + +− − . [4] 2 (a) It is given that the roots to the equation 32i 5i 0x x ax b− + + + = , where a and b are purely imaginary, are 2 i, 2 i and 1.+− Explain why the complex roots occur in conjugate pairs. [2] (b) By using (a) and an appropriate substitution, find the roots of the equation 3 2 *i 5i 0x x a x b+ + + = , where the complex conjugate of a is denoted by *a . [4] 3 (a) Given that f is a continuous function, explain, with the aid of a sketch, why the value of 2 2 4 2lim f 1 f 1 ... f 1 n n n n n n→      + + + + + +           is ( ) 3 1 f dxx . [3] (b) Hence evaluate 1 22lim ln 1 n n k k nn→ = + exactly. [4] 4 The equation of a curve C is 33 2,x xy y k+ + = where k is a constant. (a) Find d d y x in terms of x and y. [2] It is given that C has a tangent which is parallel to the y-axis. (b) Show that the y-coordinate of the points of contact of the tangent with C must satisfy 63216 4 0y y k+ + = . Hence show that 1 54k  . [6] (c) Find the possible values of k in the case where the line 6x=− is a tangent to C. [2]

3 NYJC 2025 JC2 Preliminary Examination 9758/02 [Turn Over 5 Two planes 1p and 2p have respective