EJC_9649_2025_Prelim_P2

2025 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2025 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. FURTHER MATHEMATICS Paper 2 9649/02 15 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all 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. This document consists of 6 printed pages and 2 blank pages.

2 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [50 marks] 1 Use the trapezium rule with 6 equally-spaced ordinates to approximate the value of 1.5 1 ln( 2) dxx − +∫ , giving your answer correct to 3 significant figures. With the help of a sketch, explain whether this approximation is an overestimate or underestimate. [6] 2 By finding the equation of the tangent plane to the surface 22z xy= + at 00(,)xy where 00( , ) (0,0)xy ≠ , show that (a) the tangent plane passes through the origin, and [4] (b) the tangent plane makes an angle of 45° with the z-axis. [3] 3 A circular concert hall is modelled on the coordinate plane by the region 22 4xy+≤ following an appropriate scaling of real-world distances. The deviation from ideal sound intensity at any point (,)xy on the floor of the hall is given by the function 22S( , ) 3 2xy x y x= +− where S( , )xy is measured in decibels (dB). Positive values of S indicate that the sound is louder than ideal at that point, while negative values indicate the sound is quieter than ideal. (a) Points where S( , ) 0xy = correspond to perfectly balanced acoustic spots, where the sound intensity is ideal. Find all such points that lie within the concert hall, that is, those satisfying 22 4xy+≤ . Justify your reasoning. [2] (b) Find all stationary points(s) of S( , )xy in the interior of the concert hall and determine the nature. [4] (c) Find all extreme values of S( , )xy along the boundary of the hall. Hence, determine the absolute maximum and minimum values of S( , )xy , and state where they occur. [5] 4 Let A be a 22× matri