DHS HCI RI TMJC 2025 Prelim P2

© DHS 2025 This question paper consists of 7 printed pages and 1 blank page. [Turn over 00Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2025 Year 6 FURTHER MATHEMATICS 9649/02 Paper 2 23 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Write your name, centre number, index number and class on this question paper. An answer booklet will be provided with this question paper. You should follow the instructions on the front of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Answer all the questions. 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. The number of marks is given in brackets [ ] at the end of each question or part question.

2 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Section A: Pure Mathematics [50 marks] 1 The largest positive root, , of 35e xxx −−= falls in the interval 1,N x N  + where N is an integer. (a) By sketching the graphs of 35y x x=− and e, xy −= show that 2.N = [3] (b) Use linear interpolation once on the interval  ,1NN + , with 3f ( ) 5 e xx x x −= − − to find an approximation to  , giving your answer correct to 5 decimal places. [2] The root  for the equation 35e xxx −−= satisfies the equation ( )gxx= , where ( ) ( ) 1 3g 5 e xxx −=− . An iterative method based on the form 1 g( )nnxx+ = can be used to obtain an approximation for . (c) Use the answer in part (b) as the initial approximation for  to determine the value of , correct to 4 decimal places. [3] 2 An open box is made of cardboard with the dimensions x, y, z units as shown in the diagram. The box has a fixed volume of 3 units 3. To make the box sturdy, the sides of the box labelled A and B are added with one extra layer of cardboard while the bottom of the box labelled C is added with two extra layers of cardboard. The total area of cardboard required to construct the box is denoted by S units2. You may assume that the thickness of the cardboard is negligible. (a) Find the dimensions of the box such that S is minimum. [6] (b) By differentiating S with respect to t , f ind the rate of change of S if x and y is increasing at a constant rate of 1 2 units per second and 1 3− units per second respectively when 1, 3xy== and 1z= . [3] x y z B A C

3 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination [Turn over 3 (a) Let 32M: →  be defined by ( )M a abb cc  + = . Show that M is a linear transformation. [2] (b) Let 33:T →  be a linear transformation. Let times m m T T T T=      denote the composition of the linear transformation T with itself m times. It is known that 1kkN N−  --- (*) for all positive integers k, where mN is the null space of mT . (i) Use (*) to prove that if the null space of 1kT − is equal to the null space of kT for some positive integer k, then the null space of kT is equal to the null space of 1kT + . [3] (ii) Given that m is the smallest positive integer such that ( ) mT =w0 for all 3w  and the nullity of T is non-zero, explain why 3m . [2] 4 An oncologist is studying the growth of a tumour in one of her patients. The oncologist believes the size of the tumour, P (in millimetres), at the time t months after it was first diagnosed can be modelled by the differential equation d 1,d P N P kPt =−  where k and N are constants. (a) State, in context, the significance of the constants k and N in the model. [2] (b) Given that 0P is the initial size of tumo ur when it was first diagnosed , show that an expression for the size of the tumour P after t months is 0 00 (e .) kt PN P N P −+− [4] (c) The oncologist needs to analyse the tumo ur’s growth pattern to determine the best course of action for treatment. She found that the tumour size has grown to 5 times its original size at the end of 2 months. Assuming that 025NP= , find the exact value of k. [2] After analysing the tumour growth, the oncologist decides to start her patient on chemotherapy. The growth rate of the size of tumour now follows the differential equation d 1d P kP HP P Nt = − − , where H, the therapy induced death rate, is a positive constant. (d) Given that the tumour was detected early, find the range of values of H, in terms of k, for the tumour to shrink in size and eventually be eradicated. [4]

4 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination 5 A colony of bacteria is growing on an agar plate. It is estimated that the initial population of bacteria on the plate was 6000 and that the population of bacteria at the end of the first day was 9000. From previous studies, it is known that the increase in bacteria population from day n to day ( 1)n+ is twice the increase from day ( 1)n− to day n. (a) (i) Let nb denote the population of bacteria at the end of the nth day. Find an expression for nb in terms of n. [5] (ii) Deduce the predicted size of the bacteria population at the end of the 6th day. [1] Another agar plate containing a colony of the same bacteria with same initial population was contaminated with a virus which kills the bacteria while it was trying to grow. The population of bacteria nc for this plate can be modeled by the recurrence relation 1232 1 nn n ccc k −−−= + where 0.2 1 k and nc denotes the population of bacteria at the end of nth day. (b) Given that the population of the bacteria at the end of the first day was 9000, and that the population of the bacteria was eliminated at the end of the 6th day, find the value of k. [8] Section B: Probability and Statistics [50 marks] 6 A particular machine part in a factory is known to fail over time more rapidly due to aging. Its reliability is measured in the number of hours of use before it needs to be replaced. The manufacturer of the machine part models its reliability by the random variable R (in hours) with cumulative distribution function. ( ) 3.2 32001 e , 0F 0 , 0 r rr r −  −=   (a) Find the probability that the machine part will last more than another 1400 hours given that it has lasted more than 2000 hours. [2]

5 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination [Turn over The factory owner who purchases large quantities of the machine part from the manufacturer recorded the reliability of a random sample of 210 of these parts over a period of time with the observed frequencies as follows Reliability, R (hours) Observed Frequency R < 2000 53 2000 ≤ R < 2500 41 2500 ≤ R < 3000 β 3000 ≤ R < 3500 26 3500 ≤ R < 4000 70 – β 4000 ≤ R < 4500 16 4500 ≤ R < 5000 3 R ≥ 5000 1 (b) Given that a chi-square goodness of fit test at the 5% significance level concluded that the data is not consistent with the manufacturer’s model, find the range of values of β. [5] 7 A language learning app is testing whether its new AI pronunciation coach improves users ’ pronunciation fluency. Twelve users were scored on a scale of 0 to 100 (with each score given to 1 decimal place) before and after using the app for