JPJC 9649 2023 Prelim P2

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2023 FURTHER MATHEMATICS 9649/02 Higher 2 18 September 2023 Paper 2 3 hours Additional materials: 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 booklets. If you need any 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 unsupport ed answers from a graphing calculator are not allowed in a question, you are required to pre sent 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 8 printed pages. [Turn over

2 Section A: Pure Mathematics [50 marks] 1 The sequence  nX is given by   2 1 2 n n n XX X    , for 1n . By applying th e natural logarithm to the recurrence relation, use a suitable substitution to find the general solution of the sequence, expressing your answer in trigonometric form. [5] 2 Let   1 2 1 1d n nI x x   , for 22 x   . (a) Use integration by parts to show that   12 1 2 nnn I n I  , for 0n . [3] (b) Hence, or otherwise, evaluate 3 2 I , giving your answer as a multiple of  . [3] (c) Use Simpson’s ru le with five ordinates to find a n approximation to 3 2 I , giving your answer in exact form. [2] (d) Deduce an approximation to  , giving your answer in exact form. [1]

3 3 The diagram above shows a circle with radius r cm. Initially, the lowest point P on the circumference of the circle is at the origin. As the circle rolls along the x-axis in the positive direction, P traces a path and this locus is known as a cycloid. At any time, the angle through which the rolling circle has rotated is  radians. Show that the locus of P has parametric equations    sin , 1 cosx r y r      . [2] The part of this locus for which 02  is denoted by C. (a) Sketch C and find the coordinates of its maximum point in terms of r. [3] (b) Show that the su rface area formed when C is completely rotated about the x-axis can be expressed as the integral 223 0 8 sin d 2r    , and hence, find its exact area in terms of r. [5] 4 For 1t , the function  fyt satisfies d d yt ky tt  , where k is a constant and 0y when 1t  . (a) When 1k  , find the solution for y in terms of t and k. [4] It is given that 2k  . (b) Use Euler’s method with step size 0.1 to estimate the value of y when 1.2t  . [2] (c) Sketch the curve for the solution found in (a), indicating the equation of any asymptote(s) and the coordinates of any axial intercept(s). Hence, explai n whether your answer in (b) is an over estimate or underestimate of the exact value of y when 1.2t  . [4] [Turn over P  y x O

4 5 (a) Given that the differential equation 2 2 dd 2 3 4 0dd yy yxx   is satisfied by  fyx , where  f 0 0 and  f 0 1  , determine  f x . [4] (b) Prove by induction that    3 1f 2 e sin 6 n nxx x n   for all positive integers n, where   f n x denotes  d fd n n xx . [4] (c) Sketch the graph of  fyx for 03 x  , giving the coordinates of the stationary points and the points where the graph meets the x-axis. [3] (d) The positive r oots of the equation   1f x x are den oted by 12, , ..., , ...,n   in increasing order. (i) Use the Newto n-Raphson me thod once, wi th first ap proximation 0.6, to estimate 1 . [3] (ii) State, with working, a first approximation to n when n is large. [2]

5 Section B: Probability and Statistics [50 marks] 6 The average number of bacteria in 1 ml of drugs is being investigated. (a) State, in context, an assumption that needs to be made for the number of bacteria in 1 ml of drug to be well modelled by a Poisson distribution. [1] Assume that the numbe r of bacteria in 1 ml of dr ug A has a Poiss on distribution with mean 0.5. (b) Given that the probability that there are at most 3 bacteria in k ml of drug A is less than 0.02, find the least possible integer value of k. [2] The number of the same bacteria in 1 ml of drug B has a Poisson distribution with mean 0.8. A mixture of these drugs used to treat a particular disease consists of 1. 4 ml of drug A and 1.1 ml of drug B. Bacteria in the drugs will cause infection in a patient if 5 or more bacter ia are injected. (c) Assuming that there are no chemical reactions between drugs A and B when they are mixed, find the probability that a patient will get infection after being injected with the drug mixture. [4] 7 In a particular country, large number of ducks live on lakes A and B. The mass, in kg, of a duck on lake A is denoted by x and the mass, in k g, of a duck on lak e B is denoted by y. A random sample of 8 ducks is taken from lake A and a random sample of 10 duc ks is taken from lake B. Their masses are summarised as follows. 2 2 8 10.56 14.1775 10 12.39 15.894 n x x n y y         A scientist claims that ducks on lake A are heavier on average than ducks on lake B. Test, at the 10% level of significance, whether the scientist ’s claim is justified. You shoul d assume that both distributions are normal. State another assumption necessary for the method to be valid. [7] [Turn over

6 8 Two suppliers, P and Q are under examination for the quality of their items, which are rated poor, fair or good. A random sample of items is taken from each supplier, and the numbers in each category are recorded. A test is conducted at the 5% level of significance to test whether the qua lity rating is ind ependent of the supplier and a report is generated. Unfortunately, parts of the report are dirtied by blots of ink, as shown below. Report on supplier P and Q 0H : Quality rating is independent of the supplier. 1H : Quality rating is dependent of the supplier. Data is shown as follows: Supplier Poor Fair Good Total P 120 180 Q 30 60 Total 30 60 150 240 Using a chi-squared test at the 5% level of significance, 2 6.4CALC  there is evidence to conclude that the quality rating is dependent of the supplier. (a) Calculate the expected frequencies for each category in the table found in the report. [2] (b) It is known that there are less than 20 items that are rated poor for both suppliers, find the number of items that are rated poor and fair from Supplier P and Q respectively. [4] (c) Hence, c arry out t he chi -squared test to verify if there is sufficient evidence to conclude that the quality rating is dependent of the supplier. [2]

7 9 A greengrocer has a large number of apples to sell. He claims that t he weight, X kg, of a randomly chosen apple has the probability density function given by    200 0.1 , 0.1 0.2,f 0, otherwise. xxx      (a) Find the cumulative distribution function F, of X. [2] When a customer comes to buy an apple, the greengrocer takes two apples at r andom