TJC 9758 2023 Prelim P2

1 TEMASEK JUNIOR COLLEGE 2023 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 9758/02 Paper 2 13 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, centre number and index number on the work you hand in. Write in dark blue or black pen. 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. Write your answers in the spaces provided in the Question Paper. 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 st eps 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 23 printed pages and 1 blank page. [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 Total Marks

2 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2023 9758 Section A: Pure Mathematics [40 marks] 1 (a) Prove the identity 44 311 422n n nn  + −− ≡ +     . [1] (b) The sum NS is defined by 3 1 N N n Sn = =∑ . Using the identity in part (a) find NS as a polynomial in terms of N. [5] (c) Explain why 4 NNS− converges and state the value that it converges to. [2] 2 The function f is defined by 2 2 for 0 4,f: 12 28 for 4 , xxx xx xa  + << − − ≤≤  where a is a constant. (a) State the largest value of a∈  for which the function 1f − exists. [1] For the rest of the question, let the domain of f be restricted to (0,5] . (b) Sketch the graph of f, indicating clearly the coordinates of the end point(s). [2] (c) Find 1f− in similar form. [4] The function g is defined by g( ) 1 3 for xx x += +− ∈  . (d) Show that gf exists and find its corresponding range. [2] 3 (a) By using suitable small angle approximations and standard series from the List of