JPJC 2023 JC1 Promos 9649

1 Name:____________________________________ Class: _____________ JURONG PIONEER JUNIOR COLLEGE JC1 Year End Examination 2023 FURTHER MATHEMATICS 9649/01 Higher 2 28 September 2023 Paper 1 3 hours Additional materials: List of Formulae (MF 26) READ THESE INSTRUCTIONS FIRST Write your name and civics class 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 ar e allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calcul ator are not allowed in a question, you are required to present the mathematical steps us ing mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page. [Turn over

2 1 A boy was trying to colour a 130 cm by 5 cm banne r with a crayon. He coloured an area of 125 cm2 at the end of the first hour. After each s ubsequent hour, the area he coloured is 4 5 of the area he coloured during the previous hour. (i) The area he coloured at the end of the nth hour is p cm 2. Show that ln ( ) ln 2 ( ) ln 5pA n B C n D  , where constants A, B, C and D are to be determined. [3] (ii) Show that the boy will never finish colouring the banner. [2] 2 (i) Sketch the roots of the 8th roots of unity on an Argand diagram. [2] (ii) The roots represented by 1z and 2z are such that  120a r g a r g 2zz  . Explain why the locus of all points z such that 12zz zz  passes through the origin. Draw this locus on your Argand diagram and find its exact cartesian equation. [3] 3 The terms in the sequence 012,,,uuu  satisfy the recurrence relation 12 1 2 nn nuu u  . (i) Show that the general solution of this recurrence relation is 13 13 22 nn nuc d      , where c and d are constants. [2] (ii) Find an expression for nu in terms of n in the case that 0 2u  and 1 1u  . [2] (iii) Find an expression for nu in terms of n in the case that 0 2u  and nu converges to zero. Hence find 1u . [3]

3 4 The diagram shows part of the graph of 2()yx a , where a is a positive constant, with rectangles of equal width approxima ting the area under the curve between 0x  and 1x . (i) Find the total area of the n rectangles, expressing y our answer in the form 3 1 1 f n r rn   . [2] (ii) Given that 2 1 1 (1 ) ( 21 )6 n r rn n n   , find a formula for 3 1 1 f n r rn   in terms of n and a. Hence find the area under the curve between 0x  and 1x , in terms of a. [6] 5 The complex number z satisfies the equation 32(1 2i) (1 2i) 4 0,zz z t   where t is a real number. It is give n that one root is of the form k + ki, where k is a negative real number. Find t and k, and the other roots of the equation. [9] y ......  n rectangles  x 1 O [Turn over

4 6 (a) ๠e diagram below shows a section of two polar curves with equations cos3r  and 3r  . Find the area of the shaded region. [4] (b) The arc of a curve 21yx from the point where x = 0 to the point where x = 1, is denoted by C. Show that the area of the surface generated when C is rotated through one revolution about the x-axis is  8 323 ab  , where a and b are constants to be determined. [5] 7 A special curve, known as a cycloid, is defined parametrically as follows:  sinxa t t ,  1c o sya t  , where a is a positive constant. The equations can be used to find the position of a point particle  ,Pxy travelling along a path, described by the curve, at time t seconds. (i) Express d d y x in terms of t and sketch the cycloid for 04 t  , indicating clearly the axial intercepts and stationary points. [5] (ii) Find the exact distance travelled by P along the cycloid from t  to 3t  , leaving your answer in terms of a . [5]

5 8 A medical research was done to test the effectiveness of a new antibiotic against the C-bacteria. In the beginning, there were 50 million C-bact eria. At the end of each day, the antibiotic eliminated 40% of the bacteria. In addition to the remaining bacteria, 10 million bacteria were added at the end of each day. (i) Let nC ( 0n ) denote the number of bacteria at the end of n days after the research had started. Write down a fi rst-order recurrence relation for 1nC  and solve it. [5] (ii) For this antibiotic to be considered effectiv e, it must reduce the number of bacteria to 15 million or less eventually. Using your answer to part (i), comment on the effectiveness of this antibiotic. [2] (iii) Show that the antibiotic needs to eliminate 67% of the bacteria every day in order to reduce the number of bacter ia to 15 million eventually. [3] (iv) State one assumption for this modelling process to be considered accurate. [1] 9 (i) Given that 1tan (e )xy  , find d d y x and show that 22 2 dd d 2edd d xy yy x xx   . By further differentiation of this result, find the Maclaurin series for y , up to and including the term in 3.x [6] (ii) Use your series from part (i) to estimate 0.05 1 0 tan (e ) d ,x x  correct to 5 decimal places. [2] (iii) Use your calculator to find 0.05 1 0 tan (e ) d ,x x  correct to 5 decimal places. [1] (iv) Comparing your answers to parts (ii) and (iii), and with reference to the value of x, comment on the accuracy of your approximations. [2] [Turn over

6 10 An engineer is trying to model the above triangular-wave curve mathematically. His co- worker tells him this can be done using an ‘i nfinite sum of cosine waves’. The engineer investigates possible models. (a) The first model he uses is 1 11 1cos cos 2 cos3 cos .23yx x x n x n     Sketch the graph of 1y , for 04 x  , in the case where 5n . [2] (b) The engineer decides that 1y will not give the triangular -wave curve when n . He then uses the following refinement to the above model. 2 11 1cos cos3 cos5 cos(2 1) .35 2 1yx x x n x n      Sketch the graph of 2y , for 04 x  , in the case where 5n . [2] (c) The engineer observes that 2y does not converge rapidly enough as n and so instead considers the infinite series 3 22 11 1cos cos3 cos5 cos(2 1)98 1 3 nyx x x n x        . By expressing 3y as the real part of a converge nt geometric series involving cos i sinx x , show that 3 36cos 41 9cos 2 xy x  . [8]

7 11 A conic section has polar equation 2 ,0 2sinr a    where a is a constant. (i) Given that 3a , find the equation of this curve in a standard cartesian form and show that it is an ellipse. [5] (ii) Determine the eccentricity of the curve and the coordinates of the two foci. [