VJC 9758 2025 Prelim P2

2025/VJC/Math Dept VICTORIA JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION 2025 H2 MATHEMATICS 9758/02 PAPER 2 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (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. The total number of marks for this paper is 100. This document consists of 8 printed pages.

2 2025/VJC/Math Dept Section A: Pure Mathematics [40 marks] 1 It is given that ( ) ( )f lnx a x=+ , x , xa− , where a is a constant. (a) Using the standard series from the List of Formulae (MF27), find the series expansion for ( )f x , up to and including the term in 3x . [2] It is given that 1a= . (b) Hence, or otherwise, show that the series expansion of ( )sin f x , up to and including the term in 3x is given by 2311 26 x x x−+ . [2] (c) Deduce the Maclaurin series for ( )cos f x up to and including the term in 2x . [2] (d) Find 3 1 23 d11 26 xx x x −+  . Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of ( ) 3 1 sin f d xx . [2] 2 The following diagram shows the dimensions of a trapezoidal prism with fixed volume 4 3k units3, with variables x and y. The top surface of the prism, ABCD, is an isosceles trapezoid with AB of length 5x units, DC of length 3x units, AD BC= and 60ABC BAD = = o . The rectangular sides ABFE and BCGF are perpendicular to both the top surface ABCD and the bottom surface EFGH, with AE BF DH CG y= = = = units. (a) Show that the total external surface area A of the trapezoidal prism is given by 2 3 128 kAx x+= . [4] (b) Using differentiation, find the value of x in terms of k at which A is a minimum. [4] (c) It is given instead that the volume of the prism is 1000 units3 and its external surface area is 800 units2. Find the two possible values of x. [2] A B C D E F G H 5x 3x y y y y

3 2025/VJC/Math Dept [Turn over 3 With reference to the point O as the origin and the x-y plane as a horizontal plane, t he pyramid OPQRV has a parallelogram base OPQR and height OV. The position vectors of the points P and R are 3 4 3− + + kij and 52−− kij respectively. (a) Find the coordinates of the point S that lies on the line PR such that the distance from O to S is a minimum. [3] (b) Find the cartesian equations of the planes such that the perpendicular distance from each plane to the base OPQR is 2 86 units. [3] (c) Find the acute angle between OV and the vertical. [2] (d) Given that QV is parallel to the vector 58−+ kj , find the position vector of the point V. Hence find the exact volume of the pyramid OPQRV. [5] [Volume of a pyramid = 1 base area height3 ] 4 Do not use a calculator in answering this question. The complex number z has modulus 1 and argument  , where π π2  , and the complex number w is given by i3wz= . The point P on the Argand diagram represents z. (a) On the copy of the Argand diagram with origin O in the Printed Answer Booklet, plot the points Q and R to represent w and zw− respectively. Show clearly the geometrical relationship between the points P, Q and R. [3] (b) Find the area of the quadrilateral ORPQ. [1] It is given that 3π 4 = . (c) Find z in the form ixy+ , where x and y are real numbers. [2] (d) Show that ( ) ( )3 1 3 1 iz w k − = − + + , where k is a constant to be determined. [2] (e) Hence show that 5π 3 1tan 12 31 += − . [1]

4 2025/VJC/Math Dept Section B: Probability and Statistics [60 marks] 5 A code consists of 10 characters. The first 5 characters of the code is formed using 5 letters chosen from the set {A, B, C, D, E, F, G, H} and the last 5 characters of the code is formed using 5 digits chosen from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}. The code allows for repetitions of letters and/or digits. (a) Find the number of different codes that can be formed. [1] (b) Find the probability that a code chosen at random (i) contains the letter E exactly thrice and the number 5 exactly once, [2] (ii) has E as the first character or 4 as its tenth character, but not both. [2] 6 Two events A and B are such that ( )P Ap= and ( ) 5P 4Bp= . It is given that ( ) ( )P P | 0.6A B A B== . (a) Find the value of p. [3] (b) Explain whether the events 'A and B are independent. [1] The events C and D are such that ( )P 0.55C = , ( )P 0.6D = and ( ) ( ) ( )P P PA C A D C D q= = = . (c) Find the value of q that gives the minimum value of ( )P A C D and state the minimum value of ( )P A C D . [2] 7 A shop sells apples in bags of 12. The average number of unripe apples in a bag is 2.16. (a) State two assumptions needed for the number of unripe apples in a bag to be well modelled by a binomial distribution. [2] A bag of apples is sold at a reduced price if more than 3 apples are unripe. (b) Find the probability that a randomly selected bag of apples is sold at a reduced price. [2] (c) Twenty bags of apples are selected at random. Find the probability that fewer than 4 of these bags will be sold at a reduced price. [1] (d) The shop also sells oranges in bags of n oranges. On average, the proportion of oranges that is unripe is 0.16. It is known that the modal number of unripe oranges in a bag is 2. Find the set of possible values of n. [2]

5 2025/VJC/Math Dept [Turn over 8 The diagram below shows a circular target board of radius 40 cm, divided by two concentric circles of radii 12 cm and 24 cm, into three regions, Gold, Red and Blue. Arrows are shot at the target board and the scores obtained for hitting the Gold, Red and Blue regions are 10 points, 6 points and 3 points respectively. Arrows that land outside the target board score 0 points. Alex shoots one arrow at the target board. The probability that his shot lands on the target board is p , where 0.5p . If his shot lands on the target board, the arrow is equally likely to hit any po sition on the board. Let X represent the score obtained from a single shot. You may assume that the arrow does not land on the boundary of any region. (a) Show that the ( )P 6 0.27Xp== . [1] (b) Given that ( )Var 5.464476X = , find the value of p. [4] 9 The random variable X has distribution ( ) 2N, . (a) Given that ( )P 0.45Xk= , find the value of ( )P2Xk − . [2] It is given that 4= and ( )P 1 4 0.2475X  = . (b) Find the value of  . [2] (c) Draw a sketch to show the distribution of X for x between 11− and 19 . [2]

6 2025/VJC/Math Dept 10 In this question you should state the parameters of any distributions you use. An orchard sells kiwis of two varieties, Type A and Type B. The masses, in grams, of Type A and Type B kiwis follow the distributions ( ) 2N 110, 6 and ( ) 2N 85, 8 respectively. It is assumed that these two distributions are independent. Kiwis are selected randomly and packed into bags. Type A kiwis are packed in bags of 30, while Type B kiwis are packed in bags of 6