TJC 9758 2025 Prelim P2

1 ©TJC 2025 9758/02 TEMASEK JUNIOR COLLEGE 2025 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/02 16 Sep 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all 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 must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page. [Turn over

2 ©TJC 2025 9758/02 Section A: Pure Mathematics [40 marks] 1 It is given that 3ln 1 e xy=+ . (a) Show that 23 d2e =3ed yx y x . [2] (b) By further differentiation of the result in part (a), find the Maclaurin series for y , up to and including the term in 2x , giving the coefficients in exact form. [4] (c) Deduce the Maclaurin series for 31eln 2 x− + , up to and including the term in 2x . [2] 2 (a) Using the result 2 2 1 1 ,(2 1)(2 3) 5 2 3 n r r r n= =−+ + + find 2 2 2 (2 1)(2 1) n r rr= −+ in terms of n. [4] (b) A sequence of positive numbers 1 2 3, , ,......x x x satisfies the recurrence relation 1 4 5 for 1,2,3, .nnx x n+ = + = (i) Given that the sequence converges to L, find the exact value of L. [2] (ii) Prove that 22 1 5( ).nnx L x L+ − = − [1] (iii) Use the result in part (ii) to show that if ,nxL then 1 .nxL+  [2] 3 The function f is defined by 1f : e 2 xx −+ , , 0xx . (a) Sketch the graph of ( )fyx= , ( ) 1fyx −= and ( ) 1ffyx −= on the same diagram, showing clearly their relationship. [3] The function g is defined by 2g : 1 4 , 2x x x x+ −  . (b) Explain why 1g− exists. [1] (c) Find 1g ( )x− and state its domain. [3] (d) Explain why the composite function gf exists. [2] (e) Find the exact range of gf. [2]

3 ©TJC 2025 9758/02 4 [The arc length L of a sector of radius a and angle  is given by L = a ; The volume V of a cone of base radius r and height h is given by V = 21 3 rh ] A metal sheet is shaped as a circular sector with angle  and fixed radius a, as shown in Figure 1. This sector is then formed into a right circular cone with slant height a by joining the two radii OA and OB together as shown in Figure 2. (a) If  is measured in radians, state the base radius of the cone, r, in term of a and . [1] (b) Show that the volume of the cone, V, is given by ( ) 6 2 4 2 2 4 4576 aV   =− . [3] (c) Using differentiation, find the exact value of  that will maximise V. [4] Two identical cones, as described in part (c), are joined together tip-to-tip to form a new object, as shown in Figure 3. The upper cone is initially filled with water , which leaks out into the lower cone at a rate of 31 cm s .15  − At time t seconds, the height of the water in the upper cone is y. (d) Show that the volume of water in the cone, W cm3, is given by 32 3 yW = . [2] (e) Calculate the rate of change of y when y = 2. [2] a a  A B O a r Figure 1 Figure 2 a r a r Figure 3 [Turn Over

4 ©TJC 2025 9758/02 Section B: Probability and Statistics [60 marks] 5 In a simple archery game, a player shoots an arrow at a target. The probability of hitting the target depends on the player’s skill and the wind conditions for the day. The wind can either be good or bad, with the probability of a good wind day being 0.7. According to past statistics: • Dave has a 70% chance of hitting the target on a good wind day and a 60% chance on a bad wind day. • Rafael has an 80% chance of hitting the target on a good wind day and a 50% chance on a bad wind day. Dave and Rafael decide to compete in a game where each player shoots 2 arrows at the target. The player who hits the target more times wins the game. (a) Find the probability that Dave wins the game on a good wind day. [3] It is now known that the probability Dave wins the game on a bad wind day is 0.39. (b) Given that Dave wins the game, find the probability that the day was a bad wind day. [3] 6 A treasure hunt game is to be played in a 3-storey building. There are 12 specific locations where 5 treasures are to be hidden. Among the 5 treasures, 4 of them are identical. The locations are distributed across the storeys as follows: • 3 locations on the 1st storey, • 3 locations on the 2nd storey, and • 6 locations on the 3rd storey. Each location can hold at most one treasure. (a) Find the number of ways to distribute the 5 treasures among the 12 locations for each of the following separate cases: (i) The 5 treasures can be hidden in any of the 12 locations. [1] (ii) Exactly 2 treasures must be hidden on the 3rd storey. [2] (iii) There must be at least one treasure on each of the 1st and 2nd storeys, and at least 2 treasures on the 3rd storey. [3] Five teams of 2 players each take part in the game. The 10 players and 2 game masters are to be seated around a round table for a briefing. (b) Find the number of different seating arrangements such that players from the same team must be seated next to each other but the two game masters must not be seated next to each other. [3]

5 ©TJC 2025 9758/02 7 A factory produces porcelain bowls. It is known that, on average, 8% of the bowls are faulty. The bowls are packed in boxes of 20. A box is considered imperfect if it contains more than 2 faulty bowls. Assume that the number of faulty bowls in a box follows a binomial distribution. (a) Find the probability that a randomly chosen box is imperfect. [2] These boxes are packed into cartons of 12 boxes each. (b) Find the probability that, out of 3 randomly chosen cartons, there are 2 cartons that contains fewer than 15 faulty bowls each. [3] Let Y be the number of imperfect boxes in a randomly chosen carton. (c) State the values of E(Y) and Var(Y). [2] (d) Hence using a suitable approximation, find the probability that the total number of imperfect boxes in a random sample of 35 cartons is at most 85. [2] 8 An examination consists of two parts: a written paper and a lab -based practical. Marks obtained by a randomly chosen candidate follow normal distribution with means and standard deviations as shown in the following table. Mean Standard deviation Written paper 62  Lab-based practical 56 12 It is given that the 95th percentile score for the written paper is 85 marks. (a) Show that the value of  is 13.983, correct to 5 significant figures. [1] (b) Find the probability that the difference between the marks obtained in the written paper and in the lab-based practical of a randomly chosen candidate is at least 9. [3] The overall mark obtained for the examination is the total of 60% of the mark obtained from written paper and 40% of the mark obtained from lab-based practical. (c) Find the probability that the overall mark of a randomly chosen candidate is less than 60. [3] (d) Find the smallest value of n such that the probability that the mean overall mark of n randomly chosen candidates being at leas