2023 TJC NYJC VJC H2 FM 9649 P2

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 15 September 2023 3 hours Additional Materials: Answer Booklet 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 answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Write your Civics Group and name on all the work that you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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 6 printed pages and 2 black pages. [Turn over

2 Section A: Pure Mathematics [50 Marks] 1 A particular solution to the differential equation 2d ( 1)d y yx  , has 1.6y when 0x . (a) Use the Euler method with step size 0.5 to estimate y at 1x . [2] (b) Explain with the aid of a diagram why the Euler method with the step size chosen in (b) does not provide a good approximation to the exact solution. [2] (c) Use the improved Euler method with step size 0.5 to estimate y at 1x . Suggest how the accuracy of the estimate can be improved. [3] 2 The population of a certain virus is studied in a laboratory. The research team proposes the following recurrence relation: 2 1 1nnu n au n b n n     where a and b are positive constants and un represents the size of population n days after the start of the project. Using the substitution of ,n n uv n  show that the recurrence relation can be simplified to the form 1nnv av b  . [1] (a) In the first instance, a is assigned the value 2 and u1 = 10000. (i) Find the solution for un in terms of b. [4] (ii) Determine the range of values of b for which this model predicts that population of virus will eventually be zero. For the case when 10000 b is small, describe the behaviour of un before this happens. [2] (b) If instead both a and b are assigned the value 2 and u1 = 10000, find the number of weeks such that the population exceeds 100 times of u1. [3]

3 3 Let M denote the matrix ,ab cd   where a, b, c and d are real numbers such that 1,ac 1bd and MI (I is the 22 identity matrix). (a) Show that the eigenvalues of M are 1 and 1.ad [5] (b) Find the corresponding eigenvectors of M, giving your answers where appropriate, in terms of a and d only. [4] 4 A toy company produces egg toy capsules by joining two half ellipses along their minor axes as shown in the diagram below, where a, b and c are positive constants such that a < b and a < c and b c . The toy egg capsule is obtained by rotating the region bounded by the 2 ellipses about the y- axis which is its axis of symmetry. All dimensions are in centimetres. (a) A particular type of egg capsule produced by the company is 6 cm long and 4 cm wide . Show that the volume contained by this egg capsule is 16 cubic centimetres. [4] (b) Part of the ellipse from point (0,−c) to (a, 0) is rotated about the y-axis to form the curve surface of the lower part of the capsule. This lower part of the capsule is to be coated in paint. Find the curved surface area of the capsule that is coated in paint when c = 2a. Leave your answer in terms of a. [8] x y O [Turn over

4 5 (a) Given that uy x , show that 22 2 2 2 3 d 1 d 2 d 2 d d d y u u u x x x x x x . Hence find the general solution of the differential equation 2 2 d 2 d 25 0, 0dd yy yxx x x    . [5] (b) Find the values of the constants p and q for which sin 2 cos 2y px x qx x is a particular integral of the differential equation 2 2 d 4 sin 2d y yxx  . Find the general solution of this differential equation. For the case where πxn and n is a large positive integer, show that whatever the initial conditions, y n is approximately a constant which is to be stated. [7] Section B: Probability and Statistics [50 Marks] 6 At a launch event of a new Jpad pro tablet, 150 randomly selected customers were asked to indicate their colour preference, black, white, or grey of this new model. The results, classified by gender of customers, are shown in the table below. Colour preference White Black Grey Gender Male 10 1 k 43 k Female 20 59 k 17 k (a) Given that there is no association at 5% level between gender of customers and colour preference of Jpad pro tablet. Using a chi -square test, determine the possible values of k. [6] (b) Given that 30k  , calculate the p-value. Discuss what the p-value indicates about the association between the gender of customers and colour preference of Jpad pro tablet. [2]

5 7 The stimulant Vitalin has been shown to increase the attention span and improve mathematical performance in children. A researcher selects a random sample of 10 children, conducted a suitable test on them and collected the data before and after taking the stimulant. The result s are as follows: Child 1 2 3 4 5 6 7 8 9 10 Before 15 15 13 10 8 8 6 4 5 4 After 19 12 13 11 8 13 16 17 14 2 (a) Conduct a Wilcoxon Signed Rank Test at 5% significance level to ascertain if the median score is higher after taking Vitalin. [6] (b) Show that the conclusion is reversed, if a sign test is conducted instead. Explain which conclusion should one take. [2] 8 In a computer game, a robot is placed on an infinite grid with starting position at point (0,0). The robot moves in steps and each step is either 1 unit in the positive x-direction with probability p or 1 unit in the positive y-direction with probability 1−p. The robot stops when it reaches a particular y-coordinate. (a) If the robot stops at point (X, 1), find (i)  P Xr in terms of p, [1] (ii)  P Xr in terms of p, [2] (iii) the value of p given that E(X) = 9. [2] (b) If the robot stops at point (X, 2), find in terms of p, (i)  P Xr , [1] (ii) P(X > 3). [3] 9 The continuous random variable X has probability density function 1, 0 otherw f ( ) ise, n k xx x     where n and k are constants and n is an integer greater than 3. (a) Find k in terms of n. [3] (b) Given that 1E( ) 2 nX n   , find Var(X) in terms of n, and hence prove that Var(X) < E(X). [4] (c) (i) When n = 4, find the cumulative distribution function of X. [2] (ii) Hence find P( 7| 5)XX when n = 4. [3] [Turn over

6 10 To study whether the “CGarlic diet” (Consuming 2 cloves of Cooked Garlic every day for a period of 12 weeks) can reduce the seated systolic blood pressure in working adults, a random sample of 10 adults from a large company were selected to undergo the “CGarlic diet”. The results in millimeters of mercury (mm Hg) are summarised by     2 100 149, 100 3277,xx        2 100 99, 100 1527yy    whe