DHS_HCI_RI_TMJC 2025 Prelim P2

© DHS 2025 This question paper consists of 7 printed pages and 1 blank page. [Turn over 00Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2025 Year 6 FURTHER MATHEMATICS 9649/02 Paper 2 23 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Write your name, centre number, index number and class on this question paper. An answer booklet will be provided with this question paper. You should follow the instructions on the front of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. 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. The number of marks is given in brackets [ ] at the end of each question or part question.

2 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Section A: Pure Mathematics [50 marks] 1 The largest positive root, , of 35e xxx −−= falls in the interval 1,N x N  + where N is an integer. (a) By sketching the graphs of 35y x x=− and e, xy −= show that 2.N = [3] (b) Use linear interpolation once on the interval  ,1NN + , with 3f ( ) 5 e xx x x −= − − to find an approximation to  , giving your answer correct to 5 decimal places. [2] The root  for the equation 35e xxx −−= satisfies the equation ( )gxx= , where ( ) ( ) 1 3g 5 e xxx −=− . An iterative method based on the form 1 g( )nnxx+ = can be used to obtain an approximation for . (c) Use the answer in part (b) as the initial approximation for  to determine the value of , correct to 4 decimal places. [3] 2 An open box is made of cardboard with the dimensions x, y, z units as shown in the diagram. The box has a fixed volume of 3 units 3. To make the box sturdy, the sides of the box labelled A and B are added with one extra layer of cardboard while the bottom of the box labelled C is added with two extra layers of cardboard. The total area of cardboard required to construct the box is denoted by S units2. You may assume that the thickness of the cardboard is negligible. (a) Find the dimensions of the box such that S is minimum. [6] (b) By differentiating S with respect to t , f ind the rate of change of S if x and y is increasing at a constant rate of 1 2 units per second