JPJC_9649_2023_Prelim_P1

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2023 FURTHER MATHEMATICS 9649/01 Higher 2 14 September 2023 Paper 1 3 hours Additional materials: 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 booklets. If you need any additional answer paper ask the invigilator for a continuation booklet. 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 unsupport ed answers from a graphing calculator are not allowed in a question, you are required to pre sent 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 7 printed pages and 1 blank page. [Turn over

2 1 (a) Given that A and B are square matrices with a common eigenvector x corresponding to eigenvalues  and  respectively, show that  is an eigenvalue of the matrix AB with x as a corresponding eigenvector. [2] (b) The matrix 3 1 0 466 5 11 10      A has 1 1 1   as an eigenvecto r. Find the co rresponding eigenvalue. [1] (c) The other two eigenvalues of A are 1 and 2 , with corresponding eigenvectors 1 2 3    and 1 1 2    respectively. A square matrix B has an order 3 and has eigenvalues 2, 3 and 1 with corresponding eigenve ctors 1 1 1   , 1 2 3    and 1 1 2    respectively. Find a matrix P and a diagonal matrix D such that   3 1A B PDP . [3] 2 On the first day of January 2023, a freshwater prawn farmer bought some prawns to start an aquaculture business . It is known that th e population of prawns increase s by 20% every month. O n the last day of every month, the farmer sells 40% of the prawns and bought another 14 000 prawns. Let np  1n denote the number of prawns in the farm n months after he starts his business. (a) Write down an expression for 1np  in terms of np . [2] (b) Given that the farmer bought 220 000 prawns on the fir st da y of January 2023, determine np in terms of n. [3] According to a business analyst, the business will no longer be profitable if the total number of prawns falls below 60 000. If he continues with his present business mod