RI DHS HCI TMJC 9649 2023 Prelim P2

This document consists of 8 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 [Turn over FURTHER MATHEMATICS 9649/02 Paper 2 September 2023 3 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet and a graph paper booklet will be provided with this question paper. You should follow the instructions on the front cover of both booklets. If you need additional answer paper or graph paper ask the invigilator for a continuation booklet or graph paper booklet. Write your name and CT group 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. The number of marks is given in brackets [ ] at the end of each question or part question. RAFFLES INSTITUTION 2023 YEAR 6 PRELIMINARY EXAMINATION

2 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 2 Section A: Pure Mathematics [50 marks] 1 Do not use a calculator in answering this question. (i) Use de Moivre’s theorem to prove that 35 2 4 6 6tan 20tan 6tantan 6 1 15tan 15tan tan .          [3] (ii) Show that 2tan 5 2 55 .  [4] 2 The matrices A and B are defined as follows. cos sin , with 0sin cos    A and 10 01   B . The transformations 1T and 2T from 2 to 2 are defined by 1T: xx yy           A and 2T : . xx yy           B (a) By considering cos sin xr yr             for 0r and  0,2 , describe the geometrical transformation 1 T , explaining your answer. [2] (b) Describe 2T geometrically. [1] Let matrix M be 11 322 .11 322     (c) Without using a calculator, state the smallest positive integer value of k such that kM gives the identity matrix and explain your answer. [2] (d) Given that 1 =,  C M B M find the Cartesian equations of the two lines through the origin which are invariant under the transformation given by the matrix C. [3]

3 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 2 3 Pharmacokinetics is the study of the time course of absorption, distribution, metabolism, and excretion of a medication in the human body. When a medication is taken orally, it is dissolved and absorbed into the gastrointestinal (GI) tract. It is then diff used into the bloodstream through the distribution stage and eliminated from the bloodstream by the kidney and the liver through metabolism and excretion. Let the number of units of medication present in the GI tract and the bloodstream at time t hours after an oral dose is administered be given by g and b respectively. Using this model, it is given that d 1d g gt  and d ,d b gbt  where  and  are positive constants representing the rate of elimination and distribution respectively, with . (i) By setting up a second order differential equation for b, find the general solution for b in terms of , and t. [5] In the case of the medication paracetamol, 0.2 and 0.7. When 0t , 7.6b  and d 1.12d b t  . (ii) Find b in terms of t. [2] (iii) For the effect of paracetamol in the body to be significant, the next dose needs to be administered when the amount of the medication in the bloodstream next falls below 5.69 units. How many hours later should the next dose be given for the model in part (ii)? [2] [Turn over

4 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 2 4 Let  be a constant angle 0 2  . A curve C with the equation coter  in polar coordinates is called an equiangular spiral. (i) Sketch the curve ,C for 0 . [2] (ii) Now consider the part of C where 0 2  . Find, by differentiation, the value of  at the point furthest from the half -line 2   , giving your answer in terms of  . You do not need to show that this value gives the maximum distance. [2] (iii) Find the gradient of the tangent at the point  P r,  . By using the answer to (ii) or otherwise, deduce the acute angle between the tangent at P and the line OP. [3] (iv) Find an expression, in terms of  , for the length of the arc of the curve C , for 0  , simplifying your answer. [3] 5 The function f( )yx satisfies the equation 2d cos 2d y x x xyx  and f(0) 0. The value of f( ) is to be estimated, where  is a small positive number, using 2 methods. (a) Use two steps of improved Euler method to determine an approximation to f( ) in terms of  . [6] (b) Now consider the differential equation 22d 12d2 yx x xyx    , where 0y when 0x . (i) Given that 2 0 ednx nI x x   , where 0n , show that, for integers 2,n   21 2 11 e122 n nnI n I      . [3] (ii) Solve the differential equation and obtain, in terms of  , the value of y when x  . [5] By substituting 0.1  , discuss the relative merits of the two methods employed to obtain these approximations. [2]

5 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 2 Section B: Probability and Statistics [50 marks] 6 In a particular school, the proportion of students who bring their own cutlery for their meals is p. Amy collects a large random sample of n students and calculates that a symmetric 95% confidence interval for p is  0.229, 0.371 . Bob collects a different, independent, random sample of 100 students and finds that 36 of them bring their own cutlery. Based on all 100n students in the two samples, it is desired to find a symmetric % confidence interval for p of width within 0.1. Find the largest possible value of  correct to 1 decimal place. [5] 7 The academic department at an elementary school wishes to find out whether there is a need to implement a new reading programme. The department gathers information on the level of reading activity of each child in a random sample of 205 children and used school records to categorise them according to their literacy skills. The results of this analysis are shown in the table. Level of Reading Activity High Medium Low Literacy Skills Good 25 12 7 Average 35 57 27 Poor 9 13 20 Carry out a chi -squared test for the independence of the two factors, level of reading activity and literacy skills. [6] Discuss what the test indicates about the association, if any, between the two factors, and identify any issues that the academic department should be concerned about after this analysis. You should refer to the p value for your test and to the contributions of individual cells to the test statistic. [3] [Turn over

6 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 2 8 The probability density function (pdf) of the random variable X