RI 9758/02 Prelim 2023

 RI2023   [Turn over                   CANDIDATE NAME   CLASS 23   MATHEMATICS 9758/02 Paper 2 3 hours Candidates answer on the Question Paper Additional Materials: 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. 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. Write your answers in the spaces provided in the Question Paper. Give non-exact numerical answers correct to 3 signifi cant 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 calcul ator are not allowed in a question, you are required to present the mathematical steps usi ng 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. The total number of marks for this paper is 100. FOR EXAMINER’S USE SECTION A: PURE MATHEMATICS  TOTAL  Q1 Q2 Q3 Q4 Total         9 9 11 11 40                100  SECTION B: PROBABILITY AND STATISTICS  Q5 Q6 Q7 Q8 Q9 Q10 Total  6   8  9 12 12 13 60    This document consists of 22 6 printed pages and 2 blank pages.   RAFFLES INSTITUTION Mathematics Department RAFFLES INSTITUTION 2023 YEAR 6 PRELIMINARY EXAMINATION

2 H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 2      Section A: Pure Mathematics [40 marks] 1 The curve C is defined by the parametric equations 2 ,3 2 w h e r e .xt t y t t    (a) Show algebraically that 1 4x for all values of t . [2] The line N is the normal to C at the point where 1t  . (b) Find the cartesian equation of N. [3] (c) Find the area of the finite region enclosed by C, N and 1 4x , correct to 3 decimal places. [4] 2 The terms of an arithmetic progression 1, 4, 7, 10, , are grouped into sets containing 1, 3, 5, 7,  integers, as indicated below, so that the number of integers in each set after the first is two more than the number of integers in the previous set.       1 , 4,7,10 , 13,16,19, 22, 25 , 28,31,34,37, 40, 43, 46 ,  .  (a) Find the total number of integers in the first n sets, in terms of n. [2] (b) Find the last integer in the nth set, in terms of n. Hence, find the value of k, given that the integer 2023 occurs in the kth set. [4] (c) Find the sum of all the integers in the 5th to 10th set. [3]   3 It is given that  ec o s 3xyx .  (a) Show that 2 2 dd 21 0dd yy yxx  . Hence find the first four non-zero terms of the Maclaurin expansion of ec o s 3x x . [6] (b) Verify that the same result in part (a) is obtained if the standard series expansions for ex and cosx are used. [2] (c) Show that ln (1 e cos3 )x x can be expressed as 211 7ln 2 ...28xx  . [3]

3 H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 2 [Turn over     4 With reference to the origin O, the points A and B are such that OA  a and  OB  b , where a and b are two non-zero and non-parallel vectors. (a) Show that the vector () ()  na b a a a b is perpendicular to a and is parallel to the plane OAB. [4] The vectors a and b are now given by aij k and 23 . bi j k   (b) Find a unit vector m which is perpendicular to a and is parallel to plane OAB. [2] (c) Find the coordinates of the point of intersection between the line OA and the line which passes through point B and is parallel to m. [2] (d) Hence, or otherwise, find an equation of the line which is a reflection of the line OB in the line OA. [3]   Section B: Probability and Statistics [60 marks]   5 In this question you should state the parameters of any normal distributions you use. Griffles has a large collection of small massage balls. The mass of the small massage balls, in grams, are normally distributed with a mean mass of 200 grams. The mass of 98.273% of these small massage balls is between 195 grams and 205 grams. (a) Show that the standard deviation of the mass of the small massage balls is 2.1grams,  correct to 1 decimal place. [2] Griffles has another large co llection of medium massage balls. The mass of the medium massage balls, in grams, have the distribution   2N 500,1.4 . (b) Find the probability that the total mass of 6 small massage balls exceeds twice the mass of a medium massage ball by more than 210 grams. [3] (c) State an assumption needed for your calculation in part (b). [1]

4  H2 MA 9758/2023 RI Year 6 Preliminary Examination Paper 2    6 A group of 5 boys and 3 girls sit at random at a round table. Find the number of arrangements so that (a) no 2 girls are adjacent to each other, [3] (b) all 3 girls are seated together, [3] (c) exactly 2 of the 3 girls are adjacent to each other. [2]   7 For two independent events A and B, it is given that P( )A a and P( )B b . (a) Show that A' and B' are independent events. [2] It is given that P( ) 0.7AB  and 0.5b . (b) Show that 0.4a . [2] For a third event C, it is given that P( ) 0.3C  and events A and C are independent. (c) If events B and C are mutually exclusive, find P. A' B' C' [2] (d) If events B and C are not mutually exclusive, find the greatest possible value of P, A' B' C'  showing your working clearly. [3]   8 From a school-wide survey done in 2022, it was found that the students in a certain school spend an average of 45.5 hours per week on soci al media platforms. In 2023, the school’s Student Well-being team decides to carry ou t a survey on a group of 50 randomly chosen students. (a) State what it means for a sample to be random in this context. [1] The time, x hours, spent per week on social media platforms by the 50 students are summarised below.  45.5 55x                  2 45.5 1494.74x (b) Calculate unbiased estimates of the population mean and variance of the time spent by students per week on social media platforms. [2] (c) State hypotheses that can be used to test if the mean time spent by students per week differs from 45.5 hours. Work out the critical region in this case, and use it to carry out the test at the 5% level of significance, gi ving your conclusion in the context of the question. [4] (d) Explain why it is not necessary to assume that the time spent by students per week on social media platforms follows a normal dist ribution when performing the test in part (c). [1] It is now given that the time spent by students per week on social media platforms follows a normal distribution with a variance of 25 hours2. Based on observations, the Head of Student Well-being, Ms Tan, believes that the mean time spent by students per week on social media platforms has in fact increased. (e) By carrying out the test at % level of significance using data collected from 50 randomly chosen students with a mean of 47 hours, determine th e set of values of  for w