ACJC 2023 JC1 Promos 9649

ANGLO-CHINESE JUNIOR COLLEGE JC1 PROMOTIONAL EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 4 October 2023 3 hours Additional Materials: Cover Sheet Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on 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. 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 6 printed pages. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 1 By writing cos 5 in terms of cos, find exactly all solutions of the equation 5316 20 5 1 0xx x   , stating which roots are repeated. [6] 2 Use mathematical induction to show that   d πes i n 3 2es i n 3d3 n xn x n nx xx   for all positive integers n. [6] 3 (i) Find all solutions of the equation 5 10z  . [2] Let  2 2 11p1zz z zz    . (ii) By writing  p z as a geometric series, determine the solutions of the equation p 0z  . [2] Let 1wz z . (iii) Write  p z as a quadratic expression in w, and hence find the values of w for which p 0z  , giving your answers in surd form. [3] (iv) From (ii) and (iii), determine the exact values of πcos 5 and 3πcos 5 in surd form.[3] 4 A curve 1C has the equation 223( 1) 4 12xy  . (i) Find the polar equation of 1C in the form fr   . [4] Another curve 2C with polar equation 22 c o sr  , where 02 π , intersects 1C at points P and Q. (ii) Sketch 1C and 2C on the same diagram, indicating clearly all axial intercepts. [2] (iii) Find the area of triangle OPQ, where O is the origin. [4]

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 5 A recurrence relation is given by 21 23 4nn nx xx  . (i) Find the general solution of the recurrence relation in the form   cos sinn nx RA n B n   where A and B are arbitrar