2023 TJC_NYJC_VJC FM Prelim Paper 1

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 12 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 7 printed pages and 1 blank page. [Turn over

2 1 The diagram above shows part of the curve C with equation 41 ky x  , where k is a positive constant. The region R is bounded by C, the x-axis, and the lines 1x and 3x . (a) Given that the area of R is approximately 3386 31185 when Simpson’s Rule with 5 ordinates is used, find the value of k. [4] (b) Hence find the approximate value of the volume of solid formed when the region bounded by the curve with equation 24 ky xx  , the x-axis, and the lines 1x and 3x , is rotated completely about the y-axis. [2] 2 Prove by mathematical induction 12 , for 2.n nn n  [6] 3 A curve C has polar equation 1 cos3 , 0 2 .r       (a) Sketch C and determine the cartesian equation of the tangent lines to C at the pole. [3] (b) Find in terms of , an integral for the area A enclosed by the graph and evaluate A to 3 decimal places. [2] (c) Write down the polar coordinates of the points where r is maximum and show that the distance between any two points where r is maximum is 23 . [2] 1 3 x y O R

3 4 The Happy Planet Index (HPI) is an index of human well-being and environmental impact that was introduced by the New Economics Foundation in 2006. A group of graduate students are interested to model the HPI for country A. It is estimated that at the start of 2010, the HPI for country A was 50 and