EJC 2023 JC1 Promos 9649

2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC1 Promotional Examination 2023 General Certificate of Education Advanced Level Higher 2 FURTHER MATHEMATICS Paper 1 9649/01 28 September 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 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 6 printed pages.

2 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 1 The curve C has polar equation 2s i 0 2 πn3 ,r   . (a) Sketch C, indicating clearly all key features and symmetries of the curve. [3] (b) Given that r attains a minimum value at the point with 11 π6  , find the exact cartesian equation of the tangent to C at this point. [3] 2 (a) Given that y is a function of x, find expressions for d d xyx ,  2 2 d d xyx and  3 3 d d xyx in terms of x and derivatives of y. Hence conjecture an expression for d d n n xyx in terms of x and derivatives of y, for positive integers n. [2] (b) Use mathematical induction to prove th e correctness of your conjecture. [4] (c) Hence find an expression for d ed n n xxx  in terms of x. [2] 3 Consider the system of equations 51 21 1 4 21 6 1 1 xyz xy a z xa y z     where a is a real constant. (a) Determine the values of a such that the system of equations is consistent. [3] (b) Given that infinitely many solutions exist, find the solutions of this system. [3] (c) Hence state the solutions of the system of equations 52 21 1 4 21 6 5 xyz x ya z a xa y z     where a is a constant such that infinitely many solutions exist. [2]

3 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 [Turn over 4 The curve T has equation f( )yx , where 3f( ) 3 1x xx . (a) Show that T has exactly one root  in the interval [1, 2]. [2] (b) Use one stage of the linear interpolatio n process to find an approximation,  , to the root  . Explain why this  will be an underestimate. [2