2024 JC2 H2 Math prelim - ACJC

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/01 Paper 1 20 August 2024 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on all the work you hand in. Write in dark blue or black pen. 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. Write your answers in the spaces provided in the question paper. 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. The use of an approved graphing calculator is expected, where appropriate. 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. The total number of marks for this paper is 100. This document consists of 24 printed pages and 2 blank pages. [Turn Over Question Marks 1 /4 2 /5 3 /6 4 /8 5 /9 6 /9 7 /10 8 /11 9 /12 10 /12 11 /14 Total 100 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 1 The complex numbers z and w satisfy the following equations. 2i 2 2i i 2 i w wz z z w += + = + Find z and w, giving your answers in the form of iab+ where a and b are real numbers. [4] 2 Rectangular Built-in cupboard compartment An interior designer designed a built-in cupboard for his client as shown above. The built-in cupboard of length 9 a metres, a > 0, has three equal sections and each section has a semi - elliptical hole in the centre. The designer wants to fit a hollow rectangular compartment, for storage into each of the elliptical hole. Each rectangular compartment with negligible thickness, has a length of l metres, where 2la , a height of y metres, and a fixed depth. The cross-section for part of the built-in cupboard is shown in the diagram below and the elliptical holes are modelled by the equation ( ) 2 2 for ,1f for 2 ,0 x a x ax a a x a  −   −=   and ( ) ( )f 3 fx a x+= for 99 22 aa x−   , where a is a real constant. ( )fyx= (a) Write down, in terms of l and a, the value of ( )f x when 13 2x a l=+ . [1] (b) The interior designer wishes to maximi se the rectangular compartment storage space. Show that the length of the compartment l, is 2a metres, when the space is maximised. Find also the corresponding