JPJC 2024 J2 H2 FM Prelim P2 Qn

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2024 FURTHER MATHEMATICS 9649/02 Higher 2 16 September 2024 Paper 2 3 hours Additional materials: 12-page Answer Booklet (additional 4-page Answer Booklet(s), if applicable) List of Formulae (MF 26) READ THESE INSTRUCTIONS FIRST Write your name and civics class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an 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. 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 Section A: Pure Mathematics [50 marks] 1 (a) By considering 5 0 cos dxx   , use Simpson’s rule for two strips to show that cos10  is approximately a root of 2 2 2 21 1 0.30 15 15 xx          + + + − =                    [6] (b) Hence find an approximation to cos10  , giving your answer to 4 decimal places. [1] 2 A curve C has polar equation 4 3sinr k = + , where 0 2 and is a constant, 0.kk   (i) What can be said about the value of k in each of the cases • C is a parabola, • C is an ellipse, • C is a hyperbola. [3] (ii) Given that 2k = , find the equation of this curve in a standard cartesian form. [4] (iii) Determine the eccentricity of the curve and the coordinates of the two foci. [2]

3 3 (a) (i) The function f is such that f( a)f(b) < 0 where a < b. A student concludes that the equation f(x) = 0 has at least one root in the interval (a, b). Draw a sketch to illustrate why the student could be wrong. [1] (ii) The equation 2cosec 3ln 0xx−= has a root  in the interval [3, 4]. A student uses linear interpolation once on the interval to find an approximation to . Find the approximation to  given by this method and comment on the suitability of the method in this case. [3] (b) The equation 2cosec 3ln 0xx−= also has a root  in the interval [1. 0, 2. 4]. A student finds a sequence of approximations  1 2 3, , ,...x x x to  using the recurrence relation 0 2x = and ( )1Gnnxx −= for 1n . (i) In his first attempt, the student uses ( ) 1 1G sin 3lnx x −= . Calculate 1x and explain why the student will fail when attempting to calculate 2x . [2] (ii) In his second attempt, the student uses ( ) 21 cosec3Ge x x = . Find the value of  to 3 decimal places, and demonstrate how the student could verify its correctness. [3] 4 (a) (i) The complex number z is given by cos isinz =+ , where 0 2  , show that ( ) 11cos 2 zz −=+ . [2] (ii) Prove that 0 1cos cos( 2 ) ,2 n n n k n nkk = =−  where n is an integer. [5] (iii) Hence find the exact value of 32 0 cos d   . [3] (b) Illustrate in an Argand diagram, the sets of points z for which 2 (2 i)z = + + , where  is a real parameter. [2] [Turn over

4 5 A diagram below shows an ellipse C with equation 22 22 1xy ab+= , where 0 ba , with foci 1F and 2F and eccentricity e. The point 00( , )P x y lies on C. (i) Show that the equation of the tangent at P is 00 22 1x x y y ab += . [3] The tangent at P meets the directrices at S and T and points 1D and 2D are foot of perpendiculars from P to the directrices of C. Given that 12SPD TPD  = = . (ii) Show that o 2 90PF T= . [5] (iii) Show that 2 cosPF ePT = and 12PF PF PS PT= . [3] (iv) Hence prove that 12SPF TPF = . [2] O y x P P P O y x P P P S T P P

5 Section B: Statistics [50 marks] 6 A track and field coach is interested in determining if his new training program me will improve his athletes’ 400 meter sprint time and wishes to conduct an analysis. A random sample of 10 athletes is selected. Each athlete is assessed and their sprint times both before and after his new training programme are recorded. The sprint times, in seconds, are given in the table. (a) Explain why it might not be appropriate to use a test based on the t-distribution in this situation. [1] (b) Carry out a suitable Wilcoxon test, using a 5% significance level, to investigate whether or not the new training programme appears to be effective. [5] 7 The number of delays on a railway line occurring in a randomly chosen month is denoted by D. (a) State two assumptions needed for D to be well modelled by a Poisson distribution. [2] Now assume that D can be well modelled by the distribution Po(). (b) Find in terms of , the probability that D takes one of the values 0, 2, 4 or 6 and show that the probability that D is even is ( ) 21 1 e .2 −+ [5] Athlete A B C D E F G H I J Before training 86 84 88 90 92 77 89 91 90 86 After training 80 80 78 79 92 82 88 84 92 83 [Turn over

6 8 A mall manager investigates whether the time, T hours, spent by customers shopping in the mall is greater during the holiday season than the non-holiday season. Data from a random sample of 200 customers were collected during both seasons. The results are recorded in the following table. n t 2t Holiday Season 100 341 1563 Non-holiday Season 100 257 1032 (i) Calculate a 95% confidence interval for the mean time spent by a customer during the holiday season. [3] (ii) Carry out a suitable hypothesis test, defining any symbols you use. State the p-value for the test and explain what it indicates. [5] (iii) On examining the individual time spent during the two seasons, it is clear that they are not normally distributed. Explain what implications, if any, this has for the test carried out in part (ii). [2] A statistician advises that it would have been better to measure the time spent in the mall of all 200 customers once during the holiday season and once during the non -holiday season. (iv) Explain briefly what the test procedure would have been if the statistician’s advice had been followed. Explain also why that procedure would have been better. [2]

7 9 Data from a random sample of adults were collected by a cinema company. For each person, we know the type of movie they saw and whether or not they bought snacks. The percentages of people in the various categories were as follows. Snacks No Snacks Type of movie Action 10.70% 9.80% Comedy 27.25% 22.25% Family 10.30% 9.70% Horror 4.75% 5.25% (i) Given that the sample size was 2000, carry out a chi -squared t