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