TMJC DHS HCI RI FM 2024 Prelim P1

__________________________________________________________________________________ H2 FURTHER MATHEMATICS 9649/01 Paper 1 12 SEPTEMBER 2024 3 hours Additional material: Answer Booklet List of Formulae (MF26) __________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and civics group 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. 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. 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. _________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION

2 1 The matrix A is given by 11 0 0.5 = A . (a) Form a conjecture for nA in the form 11 , for , 0 n n ab n b − + −   where a and b are real constants to be determined. [2] (b) Using mathematical induction, prove the conjecture formed in part (a). [4] 2 Let tanx = . (a) Use De Moivre’s Theorem to express tan 5 in terms of x, showing your working clearly. [4] (b) Hence, without using a calculator, find the exact value of 2 πtan 5   . [3] 3 The positive numbers nx satisfy the relation ( ) 2 1 3 1,nnxx+ =− for .n + (a) Find the values of  and , where , such that when 1x = or 1x = , the sequence remains constant. [2] (b) By considering ( ) ( ) 22 1 ,nnxx+ − or otherwise, show algebraically that if 1 ,x then the sequence increases and converges to . You may assume that the sequence converges. [5] 4 (a) Express 1 1 x− as a series, where 1.x  [1] The Fibonacci sequence is given as 1 nn+2 n+u u u=+ for , 0nn with initial conditions 0 0u = and 1 1u = . Let ( ) 0 f. r r r x u x  = = You may assume that ( )f x converges. (b) By considering ( ) ( ) ( ) 2f f f ,x x x x x−− express ( )f x in partial fractions. Hence find nu in terms of n, giving your answer in exact form. [8]

3 [Turn Over Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Further Math