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 t