NYJC_VJC_TJC_9649_2025_Prelim P2

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 9649/02 Paper 2 18th 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 9649/02 Section A: Pure Mathematics [50 marks] 1 (a) Let f and g be two differentiable functions of two variables x and y. Prove that (i) ( )fg f g∇ + =∇ +∇ , [2] (ii) ( )fg f g g f∇ =∇+∇ . [2] (b) A structure S is erected on the horizontal x-y plane such that the vertical distance H of any point P on S above the x-y plane is given by ( ) ( ) 22, 100H H xy x y= =−+ , 22 100xy+≤ where ( ),xy is a point on the x-y plane directly below P. (i) Describe geometrically the set of points of S which lie on the x-y plane. [1] (ii) Show that the directional derivative of H at the highest point of S in any direction is zero. [2] 2 Fund managers observe economic global trends when deciding on how and what to invest in. For example, in this particular dynamic investment model, 22 2 dd dd yy a bytt = −+ , for ,0ab > , where ( )yt is the capital stock (in millions of dollars) at time t (in months), a is the depreciation constant and b is reinvestment feedback constant. (a) By letting d d yx t= , show that 2d d xx ax byy = −+ . [2] (b) Use the substitution 2ux= to solve for u in terms of y . [4] (c) It is given that 0.02, 0.5ab= = . Given further that 100y= and d 0d y t = initially, find d d y t in terms of y only. You may assume d 0d y t > . [2]

3 NYJC 2025 JC2 Preliminary Examination 9649/02 [Turn Over 3 (a) Do not use graphing calculator in answer ing this part. Let ω be a cube root of unity whose imaginary part is non-zero. Show that 210 ωω++ = . [1] Hence evaluate, exactly, the expression ( ) ( )( )( ) 2452 222ω ωωω− −−− . [3] (b) The complex number z is represented by the point P in an Argand diagram. Let c be a fixed complex number and r be a rea