MI 9758 2023 Prelim P2

© Millennia Institute 9758/02/PU3/Prelim/23 2023 Preliminary Examination Pre-University 3 MATHEMATICS 9758/02 Paper 2 19 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your admission number, 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. Give your answers in the spaces provided in the Question Paper. 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 questio n. The total number of marks for this paper is 100. This document consists of 28 printed pages. Qn No. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 * Total Score Max Score 5 8 8 8 11 4 7 7 7 9 13 13 100 CANDIDATE NAME CLASS ADMISSION NUMBER

2 © Millennia Institute 9758/02/PU3/Prelim/23 Section A: Pure Mathematics [40 marks] 1 (i) It is given that 3d 2d yx x xy yx =++ , 0x≠ . Using the substitution 2y vx= , show that the differential equation can be reduced to d 1d v vx = + . [2] (ii) Given that 1y= when 1x= , solve the differential equation 3d 2d yx x xy yx =++ . [3] 2 (a) Given that ( ) 2f secxx= , find ( )f x′ and ( )f x′′ . Hence, find the Maclaurin series for ( )f x , up to and including the term in 2x . [5] (b) Using standard series from the List of Formulae (MF26), expand ( ) 4 1 ax − + as far as the term in 2x , where a is a non- zero constant. Hence find the value of a for which the coefficients of the x and 2x terms in the expansion are equal. [3] 3 (a) A sequence 234, , ,...uuu is such that 1ln 1 n nu n −=  + where 2n≥ . Find in terms of N , an expression for NS , where 234 1 ...N NNS uuu u u −=++ + + , leaving your answer in the form ( )ln faN− , where a is a positive constant. [3] (b) (i) Find the set of values of x for which the series 1 3 1 n n x x ∞ =   +∑ has a finite sum. [3] (ii) Given that 0.25x= , find the value of 1 3 1 n n x x ∞ =   +∑ . [2] 4 The curve C has equation 222 20xy+= . (i) Sketch curve C, giving the exact coordinates of any points where the curve meets the axes. [2] (ii) Show that d2 d yx xy=− . [1] (iii) It is given that point ( ),P ab is on C. The normal to C at P passes through the point (1, 0). Find the possible coordinates of P. [5]

3 © Millennia Institute 9758/02/PU3/Prelim/23 [Turn over 5 The plane p has equation 254xyz+−= , and contains the line l with equation 3 22 2 a a b λ    = −+    r , where a and b are constants and λ is a parameter. (i) Find the perpendicular distance from the origin to p. [2] (ii) Show that 2a= and 1b=− . [3] Relative to O, the point Q has position vector 473−++i jk . (iii) The points C and D are on l such that they are a distance of 110 away from Q. Find the position vectors OC   and OD   . (iv) Without performing any further calculations, explain whether the shortest distance from Q to l is greater or smaller than 110 . [1] Section B: Probability and Statistics [60 marks] 6 A game is played by choosing 2 cards from 5 identical cards numbered 1 to 5, without replacement. The score, X, is the sum of the two numbers shown on the chosen cards. (i) Determine the probability distribution of the score of this game. [2] (ii) Find ( )E X and ( )Var X . [2] 7 The probability that a student is present in school on a typical day is 0.96. In a particular school, there are 28 students in each class. The number of students in a class who are present in school on a typical school day is denoted by the random variable X. A student is considered absent if he is not present in school the whole day. Assume that X can be modelled by a binomial distribution. (i) Find the probability that all students in a class are present in school on a typical school day. [1] (ii) Find the most likely number of students who are present in school on a typical school day. [1] An investigation by the form teacher will be carried out if there are more than 2 students in the class who are not present in school on a typical school day. (iii) Find the probability that an investigation is carried out on a typical school day. [2]

4 © Millennia Institute 9758/02/PU3/Prelim/23 There are 16 classes in a level. The school leaders must be alerted if an investigation has to be carried out in at least a third of the classes in the level. (iv) Find the probability that the school leaders are alerted on a typical school day. [3] 8 For events A, B and C, it is given that ( )P 0.4A = , ( )P 0.3C = and ( )P 0.1BC∩= . It is also given that events A and B are mutually exclusive, and that events A and C are independent. (i) Find ( )P AB∩ . [1] (ii) Find ( )P| AC . [1] (iii) Given also that events B and C are independent, find ( )P ABC′′′∩∩ . [3] (iv) Given instead that events B and C are not independent, find the greatest and least possible values of ( )P B . [2] 9 A class has 12 male and 8 female students. An executive committee of 4 people comprising a chairperson, a vice-chairperson, a secretary and a treasurer is to be chosen from the class. Find the number of ways that the chairperson, the vice-chairperson, the secretary and the treasurer can be chosen from the class (i) without any restrictions, [1] (ii) such that there is at least a male student and at least a female student in the executive committee. [3] At a school event, the class is allocated a table with 10 seats. It was decided that the form teacher, executive committee and 5 pre- selected students from the class will attend the event. The 10 of them sit at random around the table. (iii) Find the probability that the executive committee members are all seated in adjacent seats and the form teacher is not seated next to an executive committee member. [3] 10 In an experiment to improve road safety, the following information was gathered about the braking distance d, measured in feet, for a car travelling at different velocities v , measured in miles per hour. v 10 20 30 40 50 60 70 80 d 27 58 97 155 212 308 399 516 (i) Draw a scatter diagram for these values, labelling the axes. [1]

5 © Millennia Institute 9758/02/PU3/Prelim/23 [Turn over (ii) By calculating the relevant product moment correlation coefficients, determine whether the rel ationship between v and d is modelled better by d av b= + or by ln d av b= + . Explain how you decide which model is better, and state the equation in this case. [4] (iii) Use your equation found in part (ii) to estimate the braking distance when the car is travelling at 64 miles per hour. Explain whether your estimate is reliable. [2] (iv) Scientists from another country wish to apply the model, but the measured velocity of the car was recorded as m kilometres per hour instead. Given that 1 kilometre 0.621 miles= , re-write your equation in part (ii) in terms of d and m so that it can be used to estimate the braking distance for a given value of m. [2]