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× matrix where 01 10 3 = A . (a) Find the eigenvalues and eigenvectors of A . [5] The sequence { }012,,, ,, nuuu u  is given by 0 1,u = 1 1,u = and 11 3 10n nnu uu+− = + for 1.n≥ Define 1 n n n u u + = v and it can be shown that 1.nn −=v Av (b) By expressing A in the diagonalised form 1−QDQ , find an expression for nu in terms of n. [4] (c) Suppose 0 1 u u ′  ′ is an eigenvector of the matrix A, explain why the resulting sequence is geometric and how this relates to the corresponding eigenvalue. [2]

3 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over 5 The diagram shows the upper-half of an ellipse 2 2 2 1yx b += where 0b> , and region R bounded by the part of the ellipse and x-axis. (a) Find, in terms of b, the volume V of the solid of revolution generated by rotating region R through 2π radians about the x-axis. [2] (b) Show that the curved surface area S of this solid of revolution is given by ( ) 1 22 0 4π 1 1db b xx +−⌠⌡ . [3] (c) Find the surface area S, in terms of b where applicable, when (i) 1b= , and [2] (ii) 1b< , using the substitution 2 sin 1 x b θ= − . [5] When 1b> , it is possible to show that 22 2 2π2π ln( 1) 1 bS b bb b = + +− − . (d) In many natural and engineered systems, from soap bubbles to biological cells, the shape that minimises surface area S for a given enclosed volume V is the most efficient. (i) On the given diagram (in the answer booklet), sketch the graph of the dimensionless isoperimetric ratio 2 3 S V , against variable length b. [2] (ii) State the specific shape of the solid of revolution when this ratio is minimised. [1] y x R -1 O 1 b

4 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 2 Section B: Probability and Statistics [50 marks] 6 The number of employees absent at a multinational company on each weekday during a particular week is tabulated as follows. The human resource manager wishes to check whether the number of absentees is independent of the day of the week. Day of the week Mon Tues Wed Thurs Fri Total Number of absentees 121 87 87 91 114 500 (a) Find the frequencies expected according to the hypothesis that the number of absentees is independent of the day of the week. [1] (b) Test at the 5% level whether the differences in the observed and expected data are significant. [4] 7 Scientists at a catfood company are considering adding a new preservative (E999) to their product. In order to test the efficacy of E999, 12 tins were chosen at random from the production line and opened. Each tin is split into two equal portions : • One portion is treated with E999 • The other portion is left untreated The 24 portions are stored under identical laboratory conditions. The lengths of time for which the contents of each portion remained safe to eat were then measured and recorded in coded units. The results are tabulated below : Tin 1 2 3 4 5 6 7 8 9 10 11 12 Length of time Treated (E999) 35 37 23 38 36 36 27 32 35 41 29 33 Untreated 34 35 26 34 35 33 25 31 31 37 30 29 (a) Making suitable assumptions, which should be stated, provide a 95% confidence interval, in terms of the coded units, for the population mean difference in safe lifetime between the treated and untreated portions. [5] (b) Using the same assumption s made in (a), t est, at the 1% significance level, whether there is any significant evidence of an increase in the mean safe lifetime as a result of the addition of E999. [4]

5 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over 8 An engineer is evaluating whether a new calibration procedure improves the accuracy of a precision instrument. For each of 15 components, the measurement error (in micrometres) is recorded before and after calibration. The data below shows the error for each component before and after calibration: Component 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Before 10.2 10.5 9.9 11.0 10.8 9.8 10.4 10.0 10.2 10.3 10.6 10.0 10.7 9.8 10.0 After 12.2 9.5 10.0 10.8 9.7 7.6 9.0 10.7 7.8 9.9 10.0 8.7 13.2 8.1 9.2 (a) Using the data from the 15 components above, perform a sign test at the 5% significance level to determine whether the new calibration procedure leads to a reduction in measurement error. [4] Let W denote the Wilcoxon test statistic, defined as the sum of the ranks of the positive differences: 1 n ii i W XR = =∑ where • iR is the rank of the i-th paired difference, • 1iX = if the difference is positive and 0iX = otherwise. (b) Assuming the null hypothesis that the differences are symmetrically distributed about zero, and that there are no ties or zero differences, show that 11E( ) ( 1) and Var( ) ( 1)(2 1).4 24W nn W nn n=+ = ++ [4] The same expectation and variance are obtained if W is defined as the sum of ranks of the negative differences. (You are given that 2 1 1 ( 1)(2 1)6 n r r nn n = =++∑ ) The study was later expanded to in