NYJC_VJC_TJC_9649_2025_Prelim P1

This document consists of 7 printed pages and 1 blank page. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2025 [Turn Over NANYANG JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 15th September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question.

2 NYJC 2025 JC2 Preliminary Examination 9649/01 1 Consider the differential equation d 36 1d 35 180 yy yhx = −−  , where h is a positive constant. (a) Given that d 0d y x ≥ for some values of y , find the range of values of h. [2] (b) For 25h= , find the equilibrium values of y , stating whether they are stable or unstable. [3] 2 The curve C has equation cos 2yx x= , 0 2x π<≤ , and has a single turning point at P. (a) Show that the x -coordinate of P is a solution of the equation 111tan 022x x − −=  . [2] (b) Show that the equation in part (a) has a root α between 0.3 and 0.5. [1] The iterative formula 1 1 11tan22 n n x x − + =   with 1 0.3x = is used to find α . (c) Find, correct to 5 decimal places, the values of 2x , 3x and 4x . [2] The diagram below shows the graphs of yx= and 111tan22y x − =  for 0x> . (d) Using the sketch in your answer booklet, show how the convergence to the root α takes place. Indicate clearly on your diagram the positions of 1x , 2x , 3x and 4x in relation to α . [2] O y x

3 NYJC 2025 JC2 Preliminary Examination 9649/01 [Turn Over 3 It is known that the equation f ( ) 0,x = where 2 ,f) 22( xxx += − has a root β . To approximate β , linear interpolation is used on the interval [ ]0,k to get an initial approximation 0x , followed by the Newton-Rhapson method. (a) If 4k = , find 0x . [1] (b) Use the Newton -Raphson method repeatedly with the initial approximation found in part (a) to determine β correct to 3 decimal places, justifying the correctness of the result obtained. [3] (c) For k β> , linear interpolation is used to obtain 0x as in part (a). Find the range of values of k such that the Newton-Raphson method would f