EJC 2023 JC1 MYE 9649

2023 JC1 H2 Further Mathematics Mid-Year Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC1 Mid-Year Examination 2023 General Certificate of Education Advanced Level Higher 2 FURTHER MATHEMATICS Paper 1 [80 marks] 9649/01 28 June 2023 2 hours 30 minutes 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 answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Answer all 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. This document consists of 5 printed pages and 1 blank page(s).

2 2023 JC1 H2 Further Mathematics Mid-Year Examination Paper 1 1 Prove by mathematical induction that 122n n  for all positive integers 3n . [6] 2 A curve E has polar equation  22 3 cos sin cos r     for 02 π . (a) Taking the polar axis as the positive x-axis, find the cartesian equation of E, leaving your answer in the form 22 9ax bxy cy , where a, b and c are constants to be determined. [3] (b) Hence or otherwise, find the exact cartesian co ordinates of the point(s) of intersection between E and the graph with polar equation cs 1 sin or   for π 4 5π 4 . [3] 3 The function f is defined by 1f( ) 2x x , ,1 .xx (a) Write down expressions for 23f() , f()x x and 4f() x in the form ax b cx d   , where a, b, c and d are integers. Hence make a conjecture for f()n x in terms of n. [3] (b) Prove your conjecture fo r all positive integers n. [5] (c) Let A be the largest subset of the real numbers such that when the domain of f is replaced with A, f()n x is defined for all positive integers n. State A. [1] 4 A curve T has polar equation 2 sin 3r  where ππ  . (a) Determine the range of values of  for which the value of r is undefined. Hence state the equations of the asymptotes of T in polar form. [3] (b) Hence sketch T, indicating clearly, in polar form, the equations of the asymptotes and any lines of symmetry, and the polar coordinates of any points where r attains stationary values. [6]

3 2023 JC1 H2 Further Mathematics Mid-Year Examination Paper 1 [Turn over 5 The general equation