JPJC 2024 J2 Prelim Math P2 Qn paper

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2024 MATHEMATICS 9758/02 Higher 2 13 September 2024 Paper 2 3 hours Candidates answer on the Question Paper. Additional materials: List of Formulae (MF 26) READ THESE INSTRUCTIONS FIRST Write your name and civics class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. This document consists of 6 printed pages. [Turn over For Candidate’s Use For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 Total Marks / 100 Answer all the questions. 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 ans wers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given by [ ] at the end of each question or part question.

2 Section A: Pure Mathematics [40 marks] 1 (a) Sketch, on the same axes, the graphs of y = 3x and y = |x2 – a2| where a > 0. [2] (b) Find the exact solution of | x2 – a2| = 3x for 0 < x < a. [2] 2 (a) Find sin cos dpx qx x , where p and q are constants such that p  q and p  q. [2] (b) Given that m  0, find sin dx mx x . [3] (c) Using the result in part (b), for all positive integers m, evaluate 0 sin dx mx x   , giving your answers in the form k m where the possible value(s) of k are to be determined. [2] 3 It is given that 1tanf ( ) e xx   , where 1tan x denotes the principal values. (i) Show that     2(1 )f ' fx x x  . [1] (ii) By further differentiation of the above result, find the Maclaurin series for f ( )x , up to and including the term in 3x . [5] (iii) Using the series in part (ii) find an approximate value of 0.5 0 f ( ) dx x , giving your answer to 4 significant figures. [1] (iv) Comment on the suitability of substituting 1x into the series in part (ii) to estimate the value of 4e  . [1] 4 (i) Show that    7 5 2 9 4 2 1 2 1 n n n n n n n       . [2] (ii) Hence find   3 9 4 2 1 N n n n n n  