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 real constant satisfying ( ) ( ) 22rcz rc z r c∗∗− +− =− where c∗ and z∗ denote the conjugates of c and z respectively. Describe the locus of P. [4] 4 The curve C with pole O has polar equation given by 1 2ekr θ = , where k is a positive constant and 02 θπ≤< . The points A and B on the curve correspond to 0θ = and θα= respectively, where 0 απ<< . The length of the arc AB is denoted by S and the area of the sector OAB is denoted by T. (a) Find S and T in terms of α and k. [7] (b) Deduce that, for large values of k, TS≈ [2] (c) Sketch the graphs of C on separate diagrams, for the cases where 10k = , 50k = , 100k = and explain how the result in part (b) can be deduced from these sketches. [3]

4 NYJC 2025 JC2 Preliminary Examination 9649/02 5 The surface S has equation ( )f,z xy= , where ( ) ( ) 222f, 4x y x y x ax y=+−+ − , and a is a real constant. (a) Find fx , fy , fxx , fyy and fxy in terms of x, y and a. [3] (b) Show that the surface S has exactly one stationary point and determine the nature of this stationary point. [5] (c) A point Q ( ),1 , 2 2kh h++ lies on the surface and is close to the point P ( )0 , 1 , 2 . Using linear approximation at the point P, show that ( )2h ak≈+ . [2] (d) The plane π has equation 0αx βy γz+ += , where , and αβ γ ( 0γ ≠ ) are real constants. It intersects the surface S in a curve C that also passes through P. It is given that the tangent line to C at P is parallel to the vector 1 2 1    . (i) Find the value of a. [2] (ii) Find the equation of π . [3] Section B: Probability and Statistics [50 marks] 6 A university lecturer wished to investigate whether there was an association between the students’ year of study and their primary study approach. A random sample of 100 students was surveyed. The results are shown in the table. Primary study approach Year 1 Year 3 Study Alone 15 30 Group Study 20 10 Online Resources 10 15 (a) Carry out a chi -square test at 1% significance level on whether there is an association between the students’ year of study and their primary study approach. You should include the contribution to the test-statistic. [5] (b) Identify the two cells which make the greatest contributions to the test statistic. Provide a possible explanation for these high contributions and explain whether this contradicts your conclusion in part (a). [3]

5 NYJC 2025 JC2 Preliminary Examination 9649/02 [Turn Over 7 An airport has a lost and found service counter for passengers who lost their belongings during their flight. The random variable Y denotes the number of reports being made by passengers who lost their belongings during an interval of one hour. (a) State a condition needed to model Y by a Poisson distribution. Explain why this condition may not hold in this context. [2] It is now assumed that Y follows a Poisson distribution with mean 4.5 per hour. (b) Find the probability that there is at least one report made during an interval of one hour. [1] (c) Show that the probability that at least one report being made in a period of t hours is given by 4.51e t−− . [1] (d) Let T be the random variable denoting the time (in hours) between two successive reports. Show that T follows an exponential distribution, stating its parameter clearly. [2] (e) Find the probability that the next report is made more than 15 minutes given that there is no report being made in the first 10 minutes. [2] 8 The continuous random variable X has probability density function given by ( ) ( ) 3 0, f 1 , xk k x x kx < ≥  =  − where 1k > . (a) Show that k = 2 and find the exact value of the expectation of X. [3] (b) Random variable W is the maximum of three independent observations of X. Find the cumulative distribution function of W and hence find ( )P4W > . [4] (c) Consecutive independent observations of W are made until the first observation that exceeds 4 is obtained. The random variable S is the total number of observations that have been made up to and including the observation exceeding 4. Find the value of P(S > E(S)). [3]

6 NYJC 2025 JC2 Preliminary Examination 9649/02 9 A research team collected exact potassium measurements in mg/kg from soil samples at 10 randomly chosen tomato farms to evaluate whether potassium level is below the recommended threshold of 250 mg/kg. The data is as follows: Farm 1 2 3 4 5 6 7 8 9 10 Potassium Level (mg/kg) 180 220 250 190 210 271 89 242 199 262 (a) Researcher A decided to carry out a sign test at the 5% significance level. Perform the test and determine the conclusion of the test. [4] (b) It is given that the potassium level for Farm 7 was incorrectly recorded in the original data table and the correct value is more than 250. Assuming a Wilcoxon signed-rank test is conducted at the 5% significance level and that: • no tied ranks exist in the absolute differences, and • the conclusion of this Wilco