RI 2023 JC1 Promos 9649

This document consists of 6 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 [Turn over FURTHER MATHEMATICS 9649 September 2023 3 hours Additional materials: List of Formulae (MF26) Writing Paper 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 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 5 PROMOTION EXAMINATION

2 H2 FM 9649/2023 RI Year 5 Promotion Examination 1 A sequence 012,,, . . .uuu is such that  56An n nuB where A and B are constants and 0n  . Given that 1 79u  and 2 949u  , find the values of A and B. [3] Prove by induction that  410 2 3 nn nu  is a multiple of 19 for all non-negative integers n. [ 5 ] 2 The equation of an ellipse is 22 22 1xy ab  where 1F and 2F are its foci. 1F is at  ,0c , 2F is at  ,0c where 0ac and 22 .ba c (a) Given that the point  ,P xy lies on the ellipse, show that 2 1 ac xPF a  and hence 12 2.PFP F a [4] (b) Given that a point  4, 3Q is on the ellipse and 12 ,QF QF k show that 2 2 Aka B aB    where A and B are constants to be determined. [4] 3 (i) Using the recurrence relation 1 1 ln( 1)3 nnxx  with 1 0.8x  , write down 2x , 5x and 8x . Determine which root of the equation 3l n (1 ) 0xx  does the sequence converge to. [2] It is given that f( ) 3 l n ( 1 )xx x    and  is very close to zero. It is known, from graphical work, that the equation f( ) 0x  has two single roots, x  and x  , where   . (ii) Explain why  is close to 0. [1] (iii) The Newton-Raphson iterative formula can be written as  1 gnn nx xx  , where     fg f' xx x . Write down g x in this case. [1] (iv) The Newton-Raphson method is used to find the value of . Use two iterations of the Newton-Raphson method, wi th initial approximation 1 0x  , to show that 2 21 6    , where terms in 3 and higher powers of  have been ignored. [4]

3 H2 FM 9649/2023 RI Year 5 Promotion Examination 4 A curve C is defined parametrically by 3(cos sin ) and 3(sin cos ).xy      Find, giving your answers in terms of . (i) the exact length of the arc PQ, where P and Q are points on the curve when 0 and 2   respectively, [4] (ii) the exact area of the surface formed when the arc PQ is rotated completely about the x-axis. [5] 5 It is given that, for 1n , 2 0 de a nx nIx x . (i) Find 1I in terms of a. [1] (ii) Show that   2 1 2 11 e22 an nnI n Ia  . [3] (iii) Hence find 5I in terms of a. [3] (iv) Find the exact volume of solid of re volution formed when the region bounded between the graph of 24e xyx  and the positive x-axis is rotated through 2 radians about the y-axis. [3] [You may assume that 2 e 0 as for any .]anaa n    6 (i) The point represented by 55 i is a vertex of an equilateral triangle. Given that the vertices all lie on the locus of 2i 5z , find the complex number represented by the vertex that lies below the real axis, giving your answer in exact cartesian form ix y . [3] (ii) The complex number z satisfies 2i 5z and 2i * 73 izz  . On an Argand diagram, sketch the region in which the point representing z can lie, and show that the exact area of the region is 1cos 10 100 B CA    where A, B and C are integers to be determined. [7] [Turn over

4 H2 FM 9649/2023 RI Year 5 Promotion Examination 7 The polar equations of the curves 1C and 2C are respectively given by 3 π,022r   , and π1s i n 3 , 0 2r    . (a) Sketch 1C and 2C and find the polar coordinates of the points where the curves intersect. [4] S is the finite region between 1C and 2C , for which 3 2r  and 1s i n 3r  . (b) Obtain the perimeter of S, giving your answer correct to 3 decimal places. [3] (c) Find exactly the area of S. [4] 8 (a) Use de Moivre’s theorem to find an expression for sin 5 in terms of power of sin. Hence find 2sin 5  in the form 5ab where a and b are to be determined. Deduce 2 2sin 5  in the form of 5cd where c and d are to be determined. [7] (b) Use de Moivre’s theorem to find an expression for 6cos  in terms of cos6 , cos 4 , and cos 2 . Hence find the exact value of 64 0 cos d    . [5] 9 (i) Lee deposits $ 30 000 into an investment account at the beginning of a year. At the end of each year, he is awarded a dividend of 6% of the amount in the account during that year, and the dividend is added to the account. He needs to pay a fixed amount of yearly management fee, which is deduc ted from his account at the end of each year. The initial deposit amount in the investment account grows by 70% after 10 years. Find the total amount of management fees Lee pays in the 10 years. [5] (ii) On the same day, Roy deposits the same initial amount into an account with another investment bank. The amount in the account at the end of the thn year is denoted by $ nu . The sequence 0 ,u 1,u 2 ,u  follows the recurrence relation 0 30000u  , 1 30000ub , 12 1 50nn nub u u  for b a positive constant and 2,nn   . (a) Find an expression for nu in terms of b and n . [6] (b) If Roy has the same amount in his account as Lee after 10 years, find the value of b correct to 5 significance figures. [1]

5 H2 FM 9649/2023 RI Year 5 Promotion Examination 10 Parabolic microphone dishes are used in Amer ican Football broadcasting to capture the sound of the players and the football. Parabolic microphone operators move along the sideline, typically following the line of scrimmage and point the long-ranging mic toward the action to capture the sound of the play. As shown in Figure 1, the device consists of a parabolic reflector that collects incoming sound waves and focuses them onto a single point at which the microphone is positioned. The microphone then converts this into an electrical signal. The sound ends up being amplified because the sound energy from a larg e area is focused on a single point. In addition, the electrical signal from the micropho ne can also be amplified. The collected sound becomes part of the televised broadcast. Figure 1 Figure 2 shows a cross-sectional view of the parabolic microphone dish, superimposed on a set of coordinate axes so that it has equation 2 4xa y , where a is a positive constant. The dish measures 32 cm in diameter at the top and 8 cm deep in the centre. Take 1 unit to represent 1 cm on both axes. The micropho ne is placed at the focus of the parabola which is denoted by M. [Turn over O x M Figure 2 y