ASRJC_FM_2023_Prelim_P1 (Questions)

FURTHER MATHEMATICS 9649/01 Paper 1 14 September 2023 3 hours Additional Materials: Answer Booklets List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklet. If you need additional answer booklet, ask the invigilator for a continuation booklet. Write your name and class on the cover page and on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a 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. This document consists of 7 printed pages and 1 blank page. ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2023 Higher 2

2 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 1 Given that a is a complex number with 3a  and 0 arg( ) , 4a  shade, on an Argand diagram, the region corresponding to the complex number z for which 23zz and 3.za Hence or otherwise, find the complex number a in cartesian form for which the maximum value of z is 2 3. [5] 2 Points F and 'F are the foci of an ellipse and point P is a point on the ellipse. The line L is the tangent to the ellipse at point P. The angles  and  are the acute angles that FP and 'FP make with the tangent respectively. Using the reflective property of ellipse, prove that    2 2' sin  bFP F P where b is the length of the semi-minor axis of the ellipse. [5] 3 (i) Using the substitution 1 nuy  , show that the differential equation   23 d 1 d 1 nyy yx x n x  where 0y can be written as 23 d 1 1 d un ux xx  . [2] (ii) For 3,n find an expression for y in terms of x. [5] 4 The curve C has polar equation given by 1sin cos 22r  , 0  . (i) Find the exact maximum value of r. [2] (ii) Sketch the curve C. [2] (iii) The point in the first quadrant where r is at its maximum is P. Find the area bounded by the curve C from 0  to P, the tangent at point P and the x-axis, leaving your answe