EJC 2026 Prelim P2

2026 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2026 General Certificate of Education Advanced Level Higher 2 FURTHER MATHEMATICS Paper 2 9649/02 14 September 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (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 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [50 Marks] 1 Solve the differential equation d cot 3dy yxx += , given that πy= when π 2x= . Leave your answer in the form ( )fyx= . [5] 2 S is the set of complex numbers z for which 5 4i 3z− −  and ( )arg π6z  . (a) On a single Argand diagram, shade the region corresponding to S. [3] (b) The complex number w is the element of S which has the largest argument. Determine the exact value of w and, correct to 2 decimal places, the value of ( )arg w . [5] 3 The surface S has equation tanz y x= , for all values of y and 2 π 2 πx−   . (a) Find an expression for the cartesian equation of PT , the tangent-plane to S at a general point P with coordinates ( )0 0 0 0, , tanx y y x . Leave your answer in the form 00ax by cz x y+ + = , where a, b and c are constants in terms of 0x and 0y . [3] (b) Determine the exact coordinates of the point Q such that QT , the tangent-plane to S at Q is parallel to the plane 0x y z+ − = . [3] (c) State the cartesian equation of OT , the tangent -plane to S at the origin O. Hence find the acute angle between the planes OT and QT . [3] 4 (a) Use de Moivre’s theorem to prove that 7 5 3 6 4 2 21 35 7tan 7 , where tan7 35 21 1t t t t tt t t − + −== − + − . [5] (b) Deduce, in trigonometric form, the roots of the equation 32 21 35 7 0z z z− + − = . [4] (c) Use the roots in part (b) to find the exact value of 2 2 2cot cot cπ ot7 2π 3π77     ++           . [4]

3 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over 5 The surface P has equation ( )f,z x y= , where ( ) ( ) ( ) 222f , 4x y k x y y x y= + + − − , and k is a real constant. (a) Find f x , f y , f xx , f yy and f xy in terms of x, y and k. [3] (b) Hence determine the values of k for which P has no stationary points. [3] (c) For each range of values of k where P has stationary points, determine the number of turning points and the nature of each of these turning points. [5] Another surface R has equation ( ) ( )( ) ( )g , sin cos 2sinz x y x y xy= = − . (d) Determine the quadratic approximation to R, ( )Q,z x y= at the point where π 4x= and 0y= . Leave all coefficients in exact form. [4] Section B: Probability and Statistics [50 Marks] 6 A World Cup Referee Coin has one face coloured black and the other face coloured yellow. Kay Long got hold of one such coin. He tossed the coin 6 times and noted the number of times he obtained black. Repeating this 80 times, he collected the following data: No. of black 0 1 2 3 4 5 6 Frequency 5 6 19 25 17 7 1 He decided to use a goodness -of-fit test to determine if the number of black in 6 tosses follows a particular probability distribution. (a) State the hypotheses for the test and define clearly any symbols used. [2] (b) Estimate the parameter needed for the test. [2] (c) Carry out the goodness-of-fit test. [4]

4 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 2 7 The Tween Tyrants is a collection of 5 different figurines sold individually in blind boxes. A buyer would only know which figurine they have bought after paying for the box and opening it. Xing has 4 different figurines in her collection and is missing only the last one. (a) State the assumptions under which the number of boxes Xing has to buy till she gets the last figurine may be well modelled by a geometric distribution. [1] It is now given that the above assumptions hold. In addition, it is assumed that each figurine is equally likely to be found in any box. (b) Find the probability that Xing has to buy at least 5 boxes to get the last figurine. [2] Yun has 3 different figurines in her collection and is missing 2 more. (c) State the most probable number of boxes Yun has to buy to get a figurine not already in her collection. [1] (d) The probability that Yun has to buy at most n boxes to get a figurine not already in her collection is more than 0.99. Find the least value of n . [3] Xing has spent more than $100 on buying the boxes in her attempt to get the last figurine. Wan has just got his 4th different figurine in the last box he bought. (e) Between Xing and Wan, who is more likely to complete their collection with the next box they buy ? Explain your answer. [2] 8 During the Perseid Meteor Shower this year, an average rate of 12 shooting stars per hour was spotted. It is assumed that the number of shooting stars spotted follows a Poisson distribution. Ryusei spent 2 hours spotting shooting stars one night during the meteor shower. (a) Find the probability that more than 24 shooting stars were spotted. [2] (b) Find the most probable number(s) of shooting stars spotted. [1] It is given that T minutes is the waiting time for the next shooting star. (c) State the probability density function of T and use it to find the probability that Ryusei has to wait more than 10 minutes for the next shooting star. [3] (d) Find the probability that Ryusei has to wait more than 10 minutes to spot a total of 2 shooting stars. [3]

5 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over 9 Cyberspell, a new mobile game company, has just installed a load balancer to direct incoming network traffic to their two servers. According to the industry standard, the mean response time for a load balancer is 250 ms. They want to test if their load balancer is meeting that standard. If it is too slow, the system is lagging; if it is too fast, it could be skipping some security checks. They did a quick check by collecting a sample of 20 response time with the following summary: 5300x= and 2 1412100x = , where x ms is a response time of their load balancer. (a) Construct a 95% confidence interval for the mean response time of the load balancer. [3] (b) State two assumptions necessary for constructing the confidence interval. [2] Instead of conducting a 5% significance test to check if the mean response time of the load balancer is 250 ms, they were able to arrive at a conclusion using the confidence interval found in (a). (c) State the conclusion and explain briefly how this is arrived at. [2] The load balancer has been set to split the traffic evenly between the two servers. To check this, they collected a random sample of 2000 user sessions and found 1060 of them being routed to Server A and the rest to Server B. (d) Construct a 95% confidence interval for the proportion of traffic directed to Server A by the load balancer. [3] Under the assumption that the population proportion of traffic