TMJC DHS HCI RI FM 2024 Prelim P2

1 [Turn over Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Further Mathematics __________________________________________________________________________________ H2 FURTHER MATHEMATICS 9649/02 Paper 2 17 SEPTEMBER 2024 3 hours Additional material: Answer Booklet List of Formulae (MF26) __________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and civics group 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 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. The total number of marks for this paper is 100. _________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION

2 Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Further Mathematics Section A: Pure Mathematics [50 marks] 1 A common logistic model to describe how population P (in thousands) varies with time t (in months) is given as d 1d PP rP htk = − − , where r, k and h are positive real constants. It is given that there are two distinct and positive equilibrium population values. (a) Find, in terms of r and k, the range of values that h can take. [2] (b) On a single diagram, sketch all possible cases of the behaviour of the population against time, as the initial population varies. You may assume that the initial population is positive. [4] 2 Adam and Bernard start with $k and $(N – k) respectively to play a game, where k and N are positive integers and .Nk Adam has a probability p, 0 1,p of winning each round. The winner of a round wins $1 from the opponent and they repeat the rounds until one player wins all the money from the opponent. kR is the probability that Adam wins all the money given that he starts with $ k and it satisfies the following recurrence relation ( )11 1,k k kR pR p R+−= + − where 0 0R = and 1.NR = (a) By considering the cases 1 2p= and 1 ,2p find the corresponding expressions of