NYJC 2023 FM CT2 P2 - modified

JC2 CT2 FM 9649/02 1 NANYANG JUNIOR COLLEGE JC2 Common Test 2 Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 4th July 2023 3 Hours Additional Materials: Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work 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, 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 5 printed pages.

Section A: Pure Mathematics [50 marks] 1 Solve the recurrence relation 21 0n n n yyy++ ++= , 0,n 01 3, 0yy= = , expressing ny as a single trigonometric function. [4] 2 Do not use a calculator in answering this question. The linear transformation 33:T →  is represented by the matrix 14 47 2 2 0 a a   − = A , where 0a . (a) Determine the value of a such that the kernel of T is a line through the origin, and hence state the equation of this line. [4] (b) Using the value of a in (a), show that the range of T is a plane through the origin and state its cartesian equation. [2] 3 Express the complex number 2i1e + where 11ππ22 −   in exponential form, justifying your working. [3] Use de Moivre’s theorem to sum the series ( ) ( ) 0 ! sin 2 1!! n r n rr n r  = +− . [6] 4 Use differentiation to prove that for all 0x , 1 2tan 1 xx x −  + and state the value of x at which equality is achieved. [4] The region in the first quadrant bounded by the curves 1tanyx −= , 21 xy x= + and the line 1x= is rotated about the y-axis through 4 right angles. Find the exact volume of the solid of revolution. [5]

5 In one model of population growth, it is assumed that the rate at which a species of fish is caught is directly proportional to the population N of the fish in the lake. With this assumption, the logistic model is replaced by the Schaefer model, given by the equation d 1d NN r N ENtM = − − where r, E and M are positive constants. (a) Give the meaning of the term EN in the above differential equation. [1] (b) Show that if Er , then there are two equilibrium populations given by 12 and N N N N== where 12NN , giving the values of 1N and 2N in terms of r, E and M where appropriate. [3] (c) With the aid of a sketch, explain why 1NN= is unstable and that 2NN= is stable. [3] The sustainable yield Y is defined to be the rate at which fish are caught when the fish population reaches stable equilibrium and E is defined to be the total effort to harvest the species of fish. (d) Express Y as a function of E and sketch the graph of Y against E, called the yield-effort curve. [2] (e) Determine E so as to maximise Y, and hence find the maximum sustainable yield mY . [2] 6 (a) The region R is bounded by the curve sinyx= , the x-axis and the line π 2x= . Using shell method, find the exact volume of the solid generated when R is rotated through 2π radians about the y-axis. [2] (b) A scientist is analysing a surface whose upper edge follows the curve sin ,yx= where 0 π.x This surface is to be created by bending a rod along the curve and rotating it through 2π radians about the x-axis to form a 3-dimensional structure. (i) Find the arc length of the upper edge, correct your answer to 4 decimal places. [2] (ii) Let A be the surface area of revolution of this flexible surface. Show that A can be expressed as 1 2 0 π 1 dk u u + , where k is a constant and u is a trigonometric expression in terms of x to be determined. [3] (iii) Hence by using the substitution tanu = , find the exact value of A. [4]

Section B: Probability and Statistics [48 marks] 7 There are two types of batteries in a bin. A battery is randomly chosen from the bin, its type noted and then placed back into the bin. The probability of choosing a type i battery with probability ip , and 2 1 1i i p = = . (a) Find the probability of drawing exactly three consecutive type 1 batteries in terms of 1p in at most six draws. [3] When in use, type i batteries last (in hours) an exponentially distributed time with rate , 1,2.i i = (b) If a randomly chosen battery is still operating after t hours of use, find the probability it will still be operating after an additional s hours. [3] 8 Independent variables and X Y are uniformly distributed over the interval from ,a x y b , where ,ab +  . (a) State P( )XY . [1] (b) By considering the cumulative distribution of max ( , )XY , show that the expected value of max( , )XY is 1 (2 )3 ba+ . [6] 9 Do not use a calculator in answering this question. The random variable X has probability density function given by 2 for ,(1 ) otherwis 2 f ( ) 0 e, kk x xx  −+=   where k is a constant. Determine the value of k and prove that 1ta| 8 2(| n)PX = . Write down the value of E( )X and find Var( )X . [9] 10 Suppose that a bookstore has customers arriving at an average of 30 customers per hour and the time between arrivals is exponentially distributed. (a) State two assumptions needed for the arrival times to be well modelled as an exponential distribution. [2] (b) After a customer arrives, find the probability that it takes less than a minute for the next custome r to arrive. [2] (c) Let A be the number of customers arriving in a 40 -minute period. State the distribution of A and hence find the probability such that A is between three standard deviations away from the mean. [3] (d) Given that the number of customers arriving in the first t minutes, ( ) 1Nt = , show that 1X , the arrival time of the first customer, is uniformly distributed in [0, t]. [4]

11 (a) In an investigation of the effects of colours on appetite, subjects were placed in either a red background themed restaurant or a blue background themed restaurant. Responses from subjects are taken and measured on some suitable scale of measurement. The researchers make the claim that “red background themed restaurants enhance the appetite in a restaurant”. For the investigation of the effect of red background themed restaurant on appetite, a sample of 20 subjects was taken, and a 95% confidence interval on the average score of appetite of subjects from the data is ( )22.28,23.29 . Find the value of the unbiased estimate o f the population variance of the score of appetite. State an assumption made in the calculation. [5] (b) Platelets are involved in the formation of blood clots, and it is known that smokers suffer more often from disorders involving blood clots than nonsmokers. To study the effect of cigarette smoking on platelet function, blood samples can be drawn from individuals before and after they smoke a cigarette and the extent to which the blood platelets come together can be measured. Dr. Peter H. Levine performed this study and published a paper, where he presented data collected from 11 individuals. We show his data in the following table, which gives the maxi