2023 TJC NYJC VJC H2 FM 9649 P2

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 15 September 2023 3 hours Additional Materials: Answer Booklet 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 answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Write your Civics Group and name on all the work that you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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 6 printed pages and 2 black pages. [Turn over

2 Section A: Pure Mathematics [50 Marks] 1 A particular solution to the differential equation 2d ( 1)d y yx  , has 1.6y when 0x . (a) Use the Euler method with step size 0.5 to estimate y at 1x . [2] (b) Explain with the aid of a diagram why the Euler method with the step size chosen in (b) does not provide a good approximation to the exact solution. [2] (c) Use the improved Euler method with step size 0.5 to estimate y at 1x . Suggest how the accuracy of the estimate can be improved. [3] 2 The population of a certain virus is studied in a laboratory. The research team proposes the following recurrence relation: 2 1 1nnu n au n b n n     where a and b are positive constants and un represents the size of population n days after the start of the project. Using the substitution of ,n n uv n  show that the recurrence relation can be simplified to the form 1nnv av b  . [1] (a) In the first instance, a is assigned the value 2 and u1 = 10000. (i) Find the solution for un in terms of b. [4] (ii) Determine the range of values of b for which this model predicts that population of virus will eventually be zero. For the case when 10000 b is small, describe the behaviour of un before this happens. [2] (b) If instead both a and b are assigned the value 2 and u1 = 10000, find the number of weeks such that the population exceeds 100 times of u1. [3]

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