MI 2026 Prelim P2 QP

©Millennia Institute 9758/02/PU3/26 2026 Preliminary Examination Pre-University 3 MATHEMATICS 9758/02 Paper 2 28 August 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must 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. This document consists of 6 printed pages. CANDIDATE NAME ADMIN NUMBER CLASS

2 ©Millennia Institute 9758/02/PU3/26 Section A: Pure Mathematics [40 marks] 1 Find the complex number z such that *( 2i) 3i 2z z   . [4] 2 The diagram below shows the graph of f ( )y x . The curve passes through the x-axis at (–2, 0) and it has a stationary point at (4, 0). The asymptotes are 0x and 2y . On separate diagrams, sketch the graphs of (i) f (2 1)y x  , [3] (ii) 1f ( )y x , [3] 3 It is given that 1 4 3 y x   . (i) Show that 1 21 31 4y x k      , where k is a constant to be found. [1] (ii) Hence, expand 1 4 3 x as a series in ascending powers of x up to and including the term in 3x . [3] (iii) Find the range of values of x for which this expansion is valid. [2] (iv) Use your answer in (ii) and integration to give an approximation for 0.8 0 1 d 4 3 x x  , giving your answer to 4 decimal places. [1] (v) Use your calculator to find numerically the value of 0.8 0 1 d 4 3 x x  , up to 4 decimal places. Why is the approximation in (iv) not very good? [2] x y

3 ©Millennia Institute 9758/02/PU3/26 [Turn over 4 (a) The non-zero vectors v and w are such that    2 3   v w v w 0 . Determine the geometrical relationship between v and w. [3] (b) It is given that a and b are non-zero vectors such that 2a and 3b . (i) If 3 a b , find the exact value of a b . [3] Referred to origin O, points A and B have position vectors a and b respectively. The point C lies on the line segment BA produced such that 3 2AC BC     . (ii) Find the position vector c of the point C in terms of a and b. [2] The point P has position vector  p a b , where  is a parameter. (iii) Find the value of  such that the points O, C and P are collinear. [3] 5 An architectural firm is designing a signature glass façade for the entrance of a museum. The cross-section of the façade is modelled as a pentagon ABCDE, as shown below. The lower portion ABDE is a rectangle with fixed vertical sides AB = DE = 10 m. The upper portion is an isosceles triangular roof BCD where BC = CD = 8 m and BCD   , where 0    . (a) Show that the total area of the glass façade, G 2m , can be expressed as 160sin 32sin2G        . [4] (b) Using calculus, (i) find the value of cos 2    that gives a stationary value of G, [4] (ii) determine whether this stationary value of G is a maximum or minimum, justifying your answer. [2] A B C E 10 10 8 8 D

4 ©Millennia Institute 9758/02/PU3/26 Section B: Probability and Statistics [60 marks] 6 A bag contains x gold coins and 2 silver coins where 1x  . In a game, a player removes coins at random from the bag without replacement, one at a time, until the player has taken out 1 gold coin. The total number of coins that the player removes from the bag is denoted by T. (i) Find, in terms of x, the probability distribution of T. [3] (ii) Given that   7E 6T  , find the value of x. [2] (iii) Hence find  Var T . [1] 7 This question is about arrangements of all nine letters in the word EXCELLENT. (i) Find the number of different arrangements that can be made in a row. [2] (ii) Given that the nine letters are arranged randomly in a row, find the probability that both ‘L’s are together and no two ‘E’s are adjacent to each other. [3] 8 On average, 6% of the audio chips produced by a manufacturer are defective. The audio chips are packed into trays, with each tray containing 50 audio chips. (i) State, in context, two assumptions needed for the number of defective audio chips in a tray to be well modelled by a binomial distribution. [2] Assume now that the number of defective audio chips in a tray has a binomial distribution. (ii) Find the probability that a randomly chosen tray of 50 audio chips contains more than 3 defective audio chips. [1] The trays are randomly packed into boxes. Each box contains 8 trays of audio chips. (iii) Find the probability that, in a randomly chosen box, at most 4 trays of audio chips contain more than 3 defective audio chips. [2] (iv) Find the probability that none of the trays in a randomly chosen box contain any defective audio chips. Give your answer in the form 10ba where a and b are constants to be determined. [2]

5 ©Millennia Institute 9758/02/PU3/26 [Turn over 9 The events A, B and C are such that  P 0.5A  ,  P 0.4B  and  P 0.15B C  . The probability that none of the three events occur is 0.1 and the events A and B are independent. (a) It is given that events A and C are mutually exclusive. (i) Write down the values of  P A B and  P A C . [2] (ii) Find  P C . [2] (iii) Find  P |B C . [2] (b) Given instead that events A and C are not mutually exclusive, and  P 0.05A B C   , find the range of possible values of  P C . [3] 10 A researcher is investigating the relationship between sleep duration and academic achievement score of university students. The average daily sleep duration, t hours, in the week before the assessments, and the academic achievement score, y points, were recorded for a random sample of 9 students. The data is given below. It is thought that this set of data can be modelled by one of the following equations: Model A: y mt c  Model B: lny a t b  (a) Give an interpretation, in context, of the value of m for Model A. [1] (b) Find the value of the product moment correlation coefficient between (i) and t y , [1] (ii) ln and t y . [1] (c) Explain whether the product moment correlation coefficient between t and y would differ if the researcher converted the average daily duration of sleep to minutes. [1] (d) By using your answers in part (b), explain whether Model A or Model B is a more appropriate model for the given set of data and find the equation of the regression line for this model. [2] (e) Use the equation of the regression line in part (d) to estimate the academic achievement score for a student who has an average daily sleep duration of 6 hours, leaving your answer to the nearest integer. Comment on the reliability of the estimate. [3] t 4.2 5.1 5.5 6.3 6.4 6.8 7.2 7.6 8.1 y 33 51 64 72 75 78 80 82 87

6 ©Millennia Institute 9758/02/PU3/26 11 A hand soap company claims that the mean volume of each bottle of their hand soap is 250 ml. The consumer association suspects that the company is overstating the mean volume of its hand soap and decides to carry out a hypothesis test on a sample of these bottles. (i) Explain why the consumer association should take a sample of at l