JPJC 2024 J2 H2 FM Prelim P1 Qn

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2024 FURTHER MATHEMATICS 9649/01 Higher 2 12 September 2024 Paper 1 3 hours Additional materials: 12-page Answer Booklet (additional 4-page Answer Booklet(s), if applicable) 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 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 6 printed pages.

2 1 Show by induction that 2 1 1 1 2 n r r n   for all integers 2n . [6] 2 The linear transformation 4 3T :   is represented by the matrix A where 1 2 2 6 1 1 2 4 3 7 6 20        A (i) Let TR and TK be the range space and null space of T respectively. Find the dimension of TR and deduce the dimension of TK . [3] (ii) By finding the bases for TR and TK , find the general solution of 3 T( ) 1 11        x . [5] 3 Marine scientists calculated that when the concentration of a particular chemical in the waters at a beach along the coast reaches 7 milligrams per litre (mg/l), the level of pollution endangers all marine life in the area. A factory wishes to release waste containing the chemical into the coastal waters. It is claimed that the discharge will not endanger the marine life in the region. e local authority is provided with the following information: - e coastal waters contains none of this chemical at present. - e factory manager has applied for a permit to discharge waste on a weekly basis into the coastal waters . e discharge, which will be done at the beginning of each week, will result in an increase in concentration of 2.5 mg/l of the chemical along the coast. - e tidal streams will remove 7% of the chemical from the coastal waters every day. (i) Based on the information, form a recurrence relation for the concentration level of chemical, nu , at the beginning of week n. Hence, find the concentration at the beginning of week