RI_DHS_HCI_TMJC_9649_2023_Prelim_P2

This document consists of 8 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 [Turn over FURTHER MATHEMATICS 9649/02 Paper 2 September 2023 3 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet and a graph paper booklet will be provided with this question paper. You should follow the instructions on the front cover of both booklets. If you need additional answer paper or graph paper ask the invigilator for a continuation booklet or graph paper booklet. Write your name and CT 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. RAFFLES INSTITUTION 2023 YEAR 6 PRELIMINARY EXAMINATION

2 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 2 Section A: Pure Mathematics [50 marks] 1 Do not use a calculator in answering this question. (i) Use de Moivre’s theorem to prove that 35 2 4 6 6tan 20tan 6tantan 6 1 15tan 15tan tan .          [3] (ii) Show that 2tan 5 2 55 .  [4] 2 The matrices A and B are defined as follows. cos sin , with 0sin cos    A and 10 01   B . The transformations 1T and 2T from 2 to 2 are defined by 1T: xx yy           A and 2T : . xx yy           B (a) By considering cos sin xr yr             for 0r and  0,2 , describe the geometrical transformation 1 T , explaining your answer. [2] (b) Describe 2T geometrically. [1] Let matrix M be 11 322 .11 322     (c) Without using a calculator, state the smallest positive integer value of k such that kM gives the identity matrix and explain your answer. [2] (d) Given that 1 =,  C M B M find the Cartesian equations of the two lines through the origin which are invariant under the transformation given by the matrix C. [3]

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