ASRJC FM 2023 Prelim P2 (Questions)

FURTHER MATHEMATICS 9649/02 Paper 2 18 September 2023 3 hours Additional Materials: Answer Booklets List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklet. If you need additional answer booklet, ask the invigilator for a continuation booklet. Write your name and class on the cover page and on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. 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. This document consists of 7 printed pages and 1 blank page. ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2023 Higher 2

2 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 1 The points P and 'P in the Argand diagram represent the complex numbers z and 'z respectively. The transformation : ',PPT where ( ) ',PPT is given by 2 *' kz z where k is a non-zero real constant and *z is the conjugate of z. Show that the line 'PP passes through the origin O. [2] Let Q represent the complex number w and ' ( ).QQ T If P and Q lie on the circles with centers at the origin and of radius 2 and 3 respectively , find the value of k for which the ratio of the area of triangle ''OP Q to area of triangle OPQ is 4 :1. Give your answer in exact form. [3] 2 Let nF be the sequence of Fibonacci numbers : 1 2 1 21, 1 and for all , 3n n nF F F F F n n       and nL be the sequence of Lucas numbers: 1 2 1 21, 3 and for all , 3n n nL L L L L n n       It can be proven that 11 for all , 1n n nL F F n n    . (a) List down the first 6 Fibonacci numbers and Lucas numbers and verify that 11 is true for 5n n nL F F n   . [2] (b) For a fixed m where 2m , prove by induction on n that 11m n m n m nF F F F F   . [5] (c) Hence or otherwise, show that 2n n nF F L . [2] 3 (a) Show that   2 21 4 sin . 2 iiee   [2] (b) For a positive integer n, series C and S are given by 2 2 21 cos cos 2 cos3 ..... cos 2 ,1 2 3 n n nCn                         2 2 2sin sin 2 sin 3 ..... sin 2 .1 2 3 n n nSn                        Show that   2 14 cos sin , 2 n nCn    and find a similar expression for S. [4] (c) Given that iwe  is a sixth root of 1, where      , find the possible values of 6(1 ) w without using a graphing calculator. [5]

3 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over 4 The terms of a sequence are denoted by 1 2 3, , ,....u u u and are related by the equation 1 1 2 4 6 , 3r r r ru u u u r      Denoting 1 2      r r r u u u by ,rx the above relation may be written as 1 M rrxx where M is a 33 matrix . Find M and verify that 1 4 9 1 , 2 , 3 1 1 1                            are eigenvectors of M. [4] Express M in the form 1PDP , where D is a diagonal matrix and P is a 33 matrix. Hence, obtain a formula for ru in terms of r, given that 1 2 3 7, 1 and 1.u u u   [7] 5 Consider the equation  f0 x  where  f sin ln , 0.x x x x   (i) Show, without the use of a graph, that there is exactly one root in the interval  0.5,1 . [2] (ii) Find an estimation for the root using linear interpolation once and determine, with clear explanation, whether it is an underestimation or overestimation of the actual root. [5] (iii) Explain why using linear interpolation a second time, with the estimated root in (ii) and 1 as the interval limits, will be unwise. [1] A student wants to use an iterative formula to find an estimate for the root of  f0 x  using 0 0.5x  as the initial estimate. (iv) By considering a necessary condition, explain why the iterative formula 1 1 1sin lnnx x     may not work. [2] (v) By using an appropriate iterative formula, find an estimate for the root, correct to 2 decimal places. Explain why, using standard series from MF26, the initial estimate of 0 0.5x  is an appropriate one. [4]

4 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over Section B: Probability and Statistics [50 marks] 6 The sine distribution, X, is defined by the probability density function given by      f sin , 0,12x x x  . (i) Find the cumulative distribution function of the sine distribution. [2] (ii) Show that, if   1 1 cos2YX  , then Y is the standard uniform distribution i.e.  0,1YU . [3] 7 Drink driving is one of the main causes of traffic accidents. Many drivers do not think that their reaction is impaired after drinking 2 cans of beer. To study whether this is true, a researcher randomly selected 20 drivers and got them to complete a reaction test before and after drinking 2 cans of beer. The results (timings in seconds) are as shown in the table below. Before 4.2 4.3 4.4 4.4 4.5 4.7 4.8 5.0 5.1 5.2 After 4.1 4.4 4.8 4.9 4.7 5.1 5.0 5.3 5.3 5.5 Before 5.3 5.4 5.8 6.0 6.1 6.1 6.3 6.5 6.6 6.6 After 5.5 5.7 5.7 5.8 6.1 6.6 6.5 6.5 6.8 6.9 Conduct an appropriate t-test to determine if drinking 2 cans of beer impairs reaction time, stating clearly a necessary assumption. [5] 8 Twelve pairs of identical twins are allocated to two training programmes, A and B, with one twin of each pair to each programme. At the end of the training the twins are observed by a panel of expert teachers and, for each pair, the better performer is determined. The results are as follows: (a) Do these observations provide evidence that programme A is a better training programme than programme B at the 5% level of significance? [4]

5 ASRJC 2023 9649/JC2 FM Preliminary Examination [Turn Over From the scoresheets of the panel of expert teachers in which the better performer is given a higher score, the differences in performance score for each pair were tabulated and the order of the pairings based on the magnitude of the absolute differences in descending order is as follows: Pair 5 9 4 3 1 8 12 2 7 11 6 10 (b) Making use of this additional information, perform another test on whether there is evidence that programme A is a better training programme than programme B at the 5% level of significance. [4] 9 An experiment was carried out to test if experienced fun claw machine players have a higher chance of ‘catching’ the prizes as compared to novice players. Two groups of 20 players each were randomly selected, with one group consisting of experienced players and the other consisting