DHS Prelims 2025 H2 Math P2 Questions

© DHS 2025 This document consists of 8 printed pages. Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9758/02 Paper 2 19 September 2025 3 hours Additional Materials : Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Write your Name, Index Number and Class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. 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 are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

2 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [40 marks] 1 A curve C has equation 9 .1yx x= + + (a) Using differentiation, find the range of values of x such that C concaves upwards. [2] (b) Sketch C. [3] 2 (a) U se the substitution ext = to show that ( ) ( ) 21 1e tan e d tan d .xx x t tt−− =∫∫ [2] (b) Hence find the exact value of ( ) 1 21 0 e tan e d .xx x− ∫ [5] 3 A curve C has equation 33 50x y xy+− = where x > 0. (a) Find d d y x in terms of x and y. [2] (b) Find the coordinates of the point on C at which the normal is parallel to the y-axis. [3] (c) Determine the nature of the stationary point. [2] 4 Relative to the origin O, the points A, B, and C have position vectors a, b and +3a b respectively. The point D lies on AB such that AD kAB= , where 0 1.k<< (a) Find a vector equation of the line OD in terms of k, a and b. [2] (b) The point E is the midpoint of BC. Find the value of k if O, D and E are collinear. [4] It is given that 1, 2 and 3 2 31.= = −=a b ab (c) By considering the scalar product (3 2 ) (3 2 ),−−abab  find the numerical value of ab and hence determine the angle between a and b. [4] (d) Give a geometrical interpretation of | |.ab [1] O A E C a b B D

3 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 [Turn over 5 A heated metal block at o160 C is left to cool in a laboratory with an ambient temperature of o30 C. The temperature, T in o C , of the