RI 2023 JC1 MYE 9649

This document consists of 4 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 FURTHER MATHEMATICS 9649 2 hours Additional materials: Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST 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 signifi cant 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 calculat or are not allowed in a question, you are required to present the mathematical steps usi ng mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, remove the cove r page and fasten it to all your work securely together. 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 65. RAFFLES INSTITUTION 2023 YEAR 5 TERM 3 COMMON TEST

2 H2 FM 9649/2023 RI Year 5 Term 3 Common Test 1 (a) Find, in terms of n, 22 11 21 n rr r    . [4] (b) Give a reason why the series 22 11 21rr r      converges, and write down its value. [2] 2 The curve C has equation 2x axy x a   , where a is a positive constant. Sketch C, indicating clearly the equations of any asymptotes, the coordinates of turning points and the coordinates of the points where the curve crosses the axes. [6] State the set of values that y can take. [1] 3 Sketch the curve C with equation 2241xy where 0y . The region R is described as the region bounded by the x-axis and C. Use differentiation to find the area of the la rgest rectangle that can be inscribed in the region R. [7] 4 (a) Obtain the expansion of 55 222(9 ) (1 3 )x x up to and including the term in 3x . Give the coefficients as exact fractions in their simplest form. [5] (b) Find the set of values of x for which the expansion in part (a) is valid. [2]

3 H2 FM 9649/2023 RI Year 5 Term 3 Common Test 5 Use de Moivre’s theorem to show that 55 3sin 5 cos ( 10 5 )tt t  where tant  . Deduce that 2tan 5  is a root of the equation 2 10 5 0xx  . Hence find the exact value of 2tan 5  . [7] 6 The complex number z satisfies 23 izz p