EJC 9649 2023 Prelim P2

2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. FURTHER MATHEMATICS Paper 2 [100 marks] 9649/02 14 September 2023 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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 6 printed pages.

2 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [50 Marks] 1 Using mathematical induction, prove that 1 3 1 1 2( 1)( 2)2 ( 2)2 N nN n n n n N     for all positive integers N. [5] Deduce the value of 1 3 ( 1)( 2)2 n n n nn     . [1] 2 (i) Let 2V  and , Vuv such that 1 2 u u  u and 1 2 v v  v . Define addition and scalar multiplication as follow: 11 22 uv uv   uv and 1 2 kuk ku  u where k is a real number. Explain why V is not a vector space with the stated operations. [2] (ii) Let M be the vector space of all real nn matrices with the standard addition and scalar multiplication for matrices. Determine if the following are subspaces of M. (a) All nn matrices A such that AB BA for a fixed nn matrix B. (b) All nn matrices A such that T A A I . [5] 3 It is given that 2 2 3dxIx   . (i) Use the trapezium rule with five ordinates to approximate I. [2] (ii) Explain whether the approximation to I found in part (i) is an under-estimate or over-estimate to I. [2] (iii) Use Simpson’s rule with five ordinates to approximate I. [2] (iv) Find the exact value of I. [2] (v) Explain clearly which of the two rules would give a more accurate approximation to I when used with the same number of ordinates. [2] (vi) Calculate the absolute percentage error in the approximation obtained in part (iii). [1]

3 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over 4 The variables x, y and z are related by the simultaneous differential equations d 4 2 8d y yzx    , (1) d 5 16d z y z xx    . (2) When 0x , 0yz . (a) By first differentiating (1), show that the simultaneous differential equations can be reduced to the second-order differential equation 2 2 dd 9 18 32 40dd yy yxxx     . [2] (b) Hence solve the differential equations to find y and z in terms of x. [10] 5 Consider the equation 42 1 3iv  . (a) Find the roots of the equation in the form ier  where 0r  and ππ   . [3] Let w denote the root with the smallest positive argument p. (b) Show that 1 2cosn nw np w for any positive integer n, and hence show that 4 7 4 3cos 16p  . [4] On an Argand diagram, the point representing the complex number w is denoted by A. (c) On the same Argand diagram, sketch the loci of points representing the complex number z given by zw and πarg( ) 3wz , indicating clearly where A is. [3] The point of intersection of the two loci is denoted by P. (d) Find the complex number represented by P and determine if it is a root of the equation in part (a). [4]

4 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 Section B: Probability and Statistics [50 marks] 6 A news agency wishes to predict the results of an election. To do so, they interview ed a random sample of voters one day before the election. 700 voters out of the sample of 900 voters said they would vote for Candidate T. (a) Find a 90% confidence interval for the proportion of the population who would vote for Candidate T. [3] (b) In order to obtain a more accurate estimate, the agency proposed to take a larger random sample. Assuming that the sample proportion remains unchanged, estimate the sample size that will be needed for the resulting 90% confidence interval to have a width of 0.04. [3] 7 A student carried out an experiment by tossing four similar coins on a table and recording how many coins landed on heads. He repeated the experiment 200 times and the results he obtained were as follows: Number of heads 0 1 2 3 4 Frequency 5 35 64 66 30 Let p be the probability that a coin will land on head. It is thought that a Binomial distribution with 0.6p would be a suitable model for the above data. Carry out a χ2-test of goodness of fit at the 5% level of significance. [4] This experiment is repeated 1000 times instead of 200 , and a similar χ2-test is performed on this new set of data. Given that the proportion for each number of heads remains the same , c omment on whether the conclusion also remains the same. [2] 8 To assess the difference in wear on the two tyres of a motorcycle, 8 motorcycles, each initially with new tyres, were ridden for 1000 km. After this time, the depth of tread on each tyre was measured and recorded. A tyre that has lesser wear has a deeper depth of tread on it. The results are shown in the table below. Motorcycle A B C D E F G H Depth of tread on front tyre (mm) 2.4 1.5 2.3 2.4 2.6 2.5 2.1 2.6 Depth of tread on rear tyre (mm) 2.3 1.9 2.1 1.8 1.8 2.8 1.4 2.1 (a) To test the hypothesis that there is no difference in the average wear for the front and rear tyres, explain why a paired sample t-test may not be appropriate. [1] (b) Test if there is a difference in the average wear for the front and rear tyres at the 1 0% level of significance, using the Wilcoxon signed rank test. [4] It was later discovered that there were two mistakes in the record. The first mistake was the depths of tread on the front and rear tyres of Motorcycle F were swapped. The second mistake was the depth of the front tyre of Motorcycle B should be b instead of 1.5. (c) Given that there was no change to the ranks, and that the Wilcoxon signed rank test at the 5% level of significance concluded that there was sufficient evidence that the front tyre has lesser average wear than the rear tyre, find the range of values of b. [2] 9 Let X be a continuous random variable with probability density function

5 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over   3f 4x x if 01 x and  f 0x  otherwise. Let 215YX . (a) Find the probability density function of Y. [6] (b) Find the expectation of Y. [2] 10 A marketing company wants to ensure it provides sufficient bandwidth to handle a certain number of the survey responses submitted on its website. The website receives an average of 20 responses per day. (i) State the conditions under which the number of responses received per day can be well modelled by a Poisson distribution. [2] Assume that these conditions in part (i) hold in a randomly chosen week. (ii) Find the probability that the marketing company received the first response in the 3 rd hour of the day. [2] (iii) Find the probability that on one day, the marketing c