TJC 2023 JC1 Promos 9649

1 TEMASEK JUNIOR COLLEGE 2023 JC1 PROMOTIONAL EXAMINATION Higher 2 FURTHER MATHEMATICS 9649 25 Sep 2023 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 both booklets. If you need 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 unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps us ing 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 5 printed pages and 3 blank pages. [Turn over

2 1 Determine whether the following sets are subspaces of 3 . Justify your answer. (a) 32 2:2 0 x Wy x y z      , (b) 3 :2 , 0 x Vy x y y z z      . [5] 2 { un} is a geometric progression with first term 2, common ratio r and sum to infinity S. Given that u1 and u3 are the thk and  th1k terms of an arithmetic progression with common difference d, show that d is negative. [3] Given further that u7 is the  th2k term of the arithmetic progression, find the exact value of the common difference. [5] 3 (a) An ellipse E is given by the equation 22 22 1xy ab , where ,ab   and a > b. F1 and F2 are the foci of E, where F1 has a positive x-coordinate and P is a point on E. (i) Explain why the perimeter of triangle F1PF2 is always a constant, and state its value in terms of a and b. [2] (ii) If F1PF2 can form an equilateral triangle, find the equations of the directrices of E in terms of a and/or b. [3] (b) The polar equation of a conic C is 3 22 s i nr   where 02   . (i) Find the eccentricity of C and identify the conic. [2] (ii) State the cartesian equa tion of the directrix of C. (You may assume that the pole is located at the origin of the cartesian plane.) [1]

3 4 A model for the population size of a particular species of bird in a particular forest is given by d 1d P PkPtN   where P is the population size (in thousands), t is the elapsed time (in number of months), and k and N are constants. (a) State the contextual significance of k and of N in this model. [2] (