MI 9758 2025 Prelim P2

©Millennia Institute 9758/02/PU3/25 2025 Preliminary Examination Pre-University 3 MATHEMATICS 9758/02 Paper 2 August/September 2025 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 8 printed pages. H CANDIDATE NAME ADMIN NUMBER CLASS

2 ©Millennia Institute 9758/02/PU3/25 Section A: Pure Mathematics [40 marks] 1 The diagram shows the curve with equation f( )yx= . The curve passes through the points P ( ), 0p , Q ( ), 0q and R ( )0, r . (a) The curve f( )yx= is transformed onto the curve with equation f (2 1)yx=− . Find the coordinates of the points on the graph of f (2 1)yx=− which correspond to the points P, Q and R on curve f( )yx= . [3] (b) It is given that f ( ) d q p I x x= . Find, in terms of I, the area of the finite region bounded by (i) the curve with equation f ( )yx=− and the x-axis, [1] (ii) the curve with equation 1 f ( 3)2yx=+ and the x-axis. [1] (c) Find the value of 0 f ( ) d q xx , justifying your answer. [1] 2 A curve C has parametric equations 2, 4, x t y t= = + where 0 16t . (i) Find the equation of the tangent to C at the point P with parameter p. [3] (ii) Given that the tangent in part (i) passes through the point (0,1) , find the exact coordinates of the point P. [3] (iii) Sketch the curve C. [2] O y x

3 ©Millennia Institute 9758/02/PU3/25 [Turn over 3 (a) The complex numbers z and w satisfy the simultaneous equations 2 * 6 iz z w+ + = + and i3zw−= . Find z and w, leaving your answers in the form iab+ , where a and b are real numbers. [4] (b) The diagram shows the curve with equation f( )yx= , where f ( )x is given by 32 5x mx nx+ + + , and m and n are real constants. The curve crosses the x-axis at the point P. One of the roots of the equation f ( ) 0x = is 2i+ . Determine the coordinates of the point P. [4] 4 A sequence is defined by 21n n nT T T++ =+ with 1 3T = , 2 1T = for 1n  . Another sequence is defined by 1n n n Tr T += for 1n  . (a) Find 4T . [1] (b) Show that 1 11n n r r + =+ for all for 1n  . [2] (c) It is given that nr converges to a limit L. Find the exact value of L. [3] A third sequence nu for 1n  , is an arithmetic sequence with first term 5 and common difference 2. (d) Find the least value of n such that 1 930. n k k u =  [2] y x O P

4 ©Millennia Institute 9758/02/PU3/25 5 The function g is given by 3g : , for , 11 xx x x x − − . (i) Sketch the graph of g( )yx= , stating clearly the coordinates of any points of intersection with the axes and the equation of any asymptotes. [3] (ii) Explain if 1g− exists. [1] (iii) Find 1g ( )x− and state the domain of 1g− . [3] A function f is said to be self-inverse if 1f ( ) f ( )xx −= for all x in the domain of f. (iv) Show that g is self-inverse. [1] (v) Without the use of a calculator, solve the equation 21g ( ) g ( )xx −=− . [2]

5 ©Millennia Institute 9758/02/PU3/25 [Turn over Section B: Probability and Statistics [60 marks] 6 The waiting times for customers to get their pastries from a popular bakery is modelled by a normal distribution with mean  minutes and standard deviation  minutes. On average, 10% of customers wait for more than 25 minutes and the same percentage of customers wait for less than 5 minutes. (i) Find the values of  and , leaving your answers to the nearest minute. [3] (ii) Explain if the normal distribution is a suitable model for the waiting times for customers to get pastries from this bakery. [1] 7 A deck of 12 cards consists of 2 gold cards and 10 silver cards. The cards are identical other than their colours. In a game, three cards are drawn from the deck of cards at random without replacement. The number of gold cards drawn is denoted by G. (i) Explain the significance of stating that the cards are identical other than their colours. [1] (ii) Show that ( ) 1P2 22G== and find the probabilities of all other possible combinations of the cards drawn. [4] A gold card is worth 4 points while a silver card is worth 1 point. The score X is the sum of the points of each of the three drawn cards. (iii) Using your answers in part (ii), find the probability distribution of X. [2] (iv) If the player of the game wins $10x for a score of x points, what is the expected winnings of the player? [1] 8 For events A and B, it is given that ( ) 11P 20A = , ( ) 4P 5AB= and ( ) 4P 11BA = . (i) Find (a) ( )P B , [3] (b) ( )P AB  . [1] (ii) Determine if the events A and B are mutually exclusive. [1] A third event C is such that ( )P 0.6C = and B and C are independent. (iii) Find ( )P BC . [1] (iv) Find the range of values of ( )P A B C . [2]

6 ©Millennia Institute 9758/02/PU3/25 9 In a particular stage of a computer game, a player is given a first set of 20 tasks, • If he completes fewer than 13 tasks, he has to retry the stage. • If he completes more than 16 tasks, he progresses directly to the next stage. • Otherwise, he is given a second set of 10 tasks. • If he completes more than 7 of these 10 tasks, he progresses to the next stage. • Otherwise, he has to retry the stage. It is known that the player’s performances in the two sets of tasks are independent. Let X and Y be the number of tasks completed in the first and second sets of tasks respectively. (i) State two assumptions needed for X to be well -modelled by a binomial distribution. [2] Assume now that X has the distribution B(20, 0.7) and Y has the distribution B(10, 0.8), (ii) Find the probability that the player is given a second set of tasks. [2] (iii) Using a tree diagram or otherwise, find the probability that the player (a) progresses to the next stage, [3] (b) completes exactly 15 tasks in the first set of tasks given he does not progress to the next stage. [2] 10 In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters. In a fruit shop, the masses, in kg, of a certain type of apple and pear are modelled as having independent normal distributions with means and standard deviations as shown in the table. Mean Standard deviation Apple 0.2 0.05 Pear 0.3 0.08 (i) An apple and a pear are chosen at random. (a) Find the probability that the mass of the apple is more than 0.25 kg and the mass of the pear is less than 0.25 kg. [1] (b) Find the probability that the mass of the apple is within 0.075 kg of the mass of the pear. [4] Apples are sold at $5 per kg and pears at $7 per kg. (ii) Find the probability that the total price of 10 randomly chosen apples and 5 randomly chosen pears exceeds $22. [4]

7 ©Millennia Institute 9758/02/PU3/25 [Turn over 11 Cardiovascular fitness is commonly measured using VO 2 ma