ACJC 2025 Prelim P2

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 18 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 are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. __________________________________________________________________________________ This document consists of 8 printed pages. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 Section A: Pure Mathematics [50 marks] 1 A small town has a population that changes every year due to two factors: • natural decline or growth, where the population becomes a times its previous size , where 0a ; • migration, where the population changes by k people because of people move in and out of the town. Let nx represent the population in year n. The population follows the recurrence relation 1−=+nnx ax k , where 2n and 1 10000.=x (i) Solve the recurrence relation to find nx in terms of ,na and .k [4] (ii) Given that 01 a , find the range of values of k such that the population does not grow indefinitely. [2] (iii) If 0.6=a and 100=k , find the value at which the population stabilizes after a long period of time. [1] 2 Let ( ) 1 1f( ) tan e xx x −=− . Show that the equation f( ) 0x = has only one root  in the interval [0.7, 1]. [2] Two iterative formulae are given below: (I) 1 1ln tann n x x +  =    ; (II) ( ) 1 1 1 tan e nn xx + −= . Choose one of the iterative formulae to find an approximation to  , and explain your choice. Using an initial approximation of 0 0.75x = for the chosen iterative formula, find an approximation to the root  , correct to three decimal places. [6]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 3 In an Argand diagram, the point P represents the complex number z that satisfies ( )( )3 i * 3 i 4zz− − − +  and 3izz − − . (i) Sketch the locus of P . [3] (ii) Find the exact maximum and minimum values of 3iz+− . [2] (iii) Find the exact complex number z such that ( )arg 3 iz+− is (a) maximum; (b) minimum, giving your answer in the form ixy+ . [5] 4 The surface S has equation ( )f , z x y= where ( ) 22 22 f , exy xyxy − −= . (a) Find the set of critical points of the surface S. [4] The point A denotes the point on S where 1x= and 2y= . (b) Find the equation of the tangent plane of S at A in exact form. [3] (c) Find the exact directional derivative of S at A in the direction of 5 12   . [3]

4 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 5 A parachute’s stabilisation system responds to brief vertical gusts during descent. The small vertical displacement y, in metres, from its steady descent path is modelled by the differential equation 2 2 2dd 2 25 5edd tyy ay tt −+ + = where t is time in seconds. (i) If 05 a and 29 4a , find the general solution of the differential equation in terms of a. [6] (ii) At time 0t = , the parachute has no vertical displacement and is stationary in the vertical direction. Assume now that 1a= . Express y in terms of t. Sketch the solution curve and find the largest vertical displacement. [5] (iii) The actual largest vertical displacement is measured to be 0.12 metres. Does this suggest that the actual value of a is greater than or less than 1? Justify your answer using your result from part (ii). [2] (iv) Suppose the differential equation is now changed to 2 2 5dd 10 25 5edd tyy ytt −+ + = . Explain why a particular integral of the form 5e tB − or 5e tCt − cannot be used to solve the differential equation. [2]

5 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 [Turn Over Section B: Statistics [50 marks] 6 A company conducted a survey with a random sample of 300 customers to investigate whether the type of advertisement shown to customers is associated with their decision to purchase a product. Three types of advertisements were released for a certain produc t: a social media ad, an email ad and a billboard ad. Customers were asked to state the type of the first advertisement they saw and whether they purchased the product or not. The results are shown in the table below. Purchased Did not purchase Total Social media 53 53 106 Email 20 84 104 Billboard 20 70 90 (a) Perform a chi-squared test for independence at the 5% significance level to determine whether there is an association between the type of advertisement and the purchase decision. [5] (b) By referring to the contributions to the test statistic, suggest with a suitable reason which advertisement method the company should invest more to maximise their sales. [2]

6 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 7 A random sample of 12 participants took part in a study to find out whether drinking regular coffee influences one’s reaction time compared to drinking decaffeinated coffee. Each of them did a reaction test after drinking decaffeinated coffee on one day an d regular coffee a week later, and their reaction times were measured on both days. The table below shows their reaction times in milliseconds, with smaller times indicating faster reactions. Participant Decaffeinated Regular A 245 220 B 243 225 C 225 230 D 248 218 E 244 222 F 226 226 G 240 228 H 227 229 I 239 224 J 241 221 K 235 227 L 239 229 (a) Carry out a suitable Wilcoxon signed rank test at 1% level of significance to investigate whether there is evidence that drinking regular coffee results in faster reaction times compared to decaffeinated coffee. [5] (b) Calculate the p-value obtained if a sign test is used instead and state if it would lead to the same conclusion. Give one reason why the Wilcoxon signed rank test is more appropriate than the sign test in this study. [2]

7 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 8 A manufacturer inspects rolls of fabric for defects. Each roll of fabric is 50 metres long. It is known from the manufacturer’s experience that the number of defects on the fabric follows a Poisson distribution with a mean of 2 defects for every 10 metres of fabric. (a) State two assumptions needed for the number of defects in a roll of fabric to be well modelled by a Poisson distribution. [2] (b) Find the probability that a randomly chosen roll of fabric contains 5 defects. [2] (c) By considering the value of 1k k p p + , where kp is the probability that a randomly chosen roll of fabric has k defects, show algebraically that the most likely numbers of defects that a roll of fabric can have is 9 and 10. [3] (d) State the distribution of the length of fabric between consecutive defects and write down the mean length in metres between consecutive defects. [2] A roll of fabric passe