ACJC 2025 Prelim P2

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 18 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 8 printed pages. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 Section A: Pure Mathematics [50 marks] 1 A small town has a population that changes every year due to two factors: • natural decline or growth, where the population becomes a times its previous size , where 0a ; • migration, where the population changes by k people because of people move in and out of the town. Let nx represent the population in year n. The population follows the recurrence relation 1−=+nnx ax k , where 2n and 1 10000.=x (i) Solve the recurrence relation to find nx in terms of ,na and .k [4] (ii) Given that 01 a , find the range of values of k such that the population does not grow indefinitely. [2] (iii) If 0.6=a and 100=k , find the value at which the population stabilizes after a long period of time. [1] 2 Let ( ) 1 1f( ) tan e xx x −=− . Show that the equation f( ) 0x = has only one root  in the interval [0.7, 1]. [2] Two iterative formulae are given below: (I) 1 1ln tann n x x +  =    ; (II) ( ) 1 1 1 tan e nn xx + −= . Choose one of the iterative formulae to find an approximation to  , and explain your choice. Using an initial approximation of 0 0.75x = for the chosen iterative formula, find an approximation to the root  , correct to three decimal places. [6]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 3 In an Argand diagram, the point P represents the complex number z that satisfies ( )( )3 i * 3 i 4zz− − − +  and 3izz − − . (i) Sketch the locus of P .