NYJC 2023 JC1 Promos 9649

This document consists of 6 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2023 [Turn Over NANYANG JUNIOR COLLEGE JC1 End of Year Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 29th September 2023 3 Hours Additional Materials: Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. 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 answers 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

2 NYJC 2023 JC1 End-of-Year Examination 9649/01 [Turn Over 1 A nn matrix A is said to be idempotent if 2 AA . (a) Show that det( )A is 0 or 1. [2] (b) Show that if det( ) 1A , then AI . [1] Let wx yz  A be a 22 idempotent matrix where 0w , 0z . (c) If det( ) 0A , find the value of wz . [2] 2 Let  fs i n 2x x . Prove by mathematical induction that   2 2 d f4 s i n 2d n n n x xx  for .n   [5] Hence, explain why the Maclaurin expansion of f(x) has only odd powers of x. [2] 3 (a) The curve C is defined parametrically by cos ln tan 2 txa t   , sinya t where 34 t  . and a is a positive constant. Find the exact length of the arc of C, in terms of a. [4] (b) Let 2 1 0 d 4 n nxIx x     , where n is a non-negative integer. Prove that for 2n ,   241 3nnnI n I   . [3] Let R be the region bounded by the curve 3 24 xy x   , the line 1 3 y and the y-axis. Find (i) the exact area of R, [3] (ii) the exact volume of the solid obtained when R is rotated 2 radians about the y-axis. [4]

3 NYJC 2023 JC1 End-of-Year Examination 9649/01 [Turn Over 4 (a) The sequence of positive numbers nu for 1, 2, 3,n Κ satisfy the recurrence relation 1 n n n ku au uk    , where a and k are positive real constants. (i) Given that the sequence converges, find its limit in terms of a. [2] (ii) If nua , by considering 1nnuu  , show that 1nnuu  . [2] (b) Two sequences     and nnx y are related by 11 110.8 0.1 and 0.2 0.9nn n nn nx xy y xy  