EJC 9758 2023 Prelim P2

2023 JC2 H2 Mathematics Preliminary Examination Paper 2 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 2 [100 marks] 9758/02 21 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on 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. DO NOT WRITE ON ANY BARCODES. Answer all 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. 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. This document consists of 24 printed pages and 4 blank pages. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total

2 2023 JC2 H2 Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics (40 marks) 1 The complex numbers z and w satisfy the following equations. 4 5i 7 (1 i) 8 30 zw zw += −+= Find z and w, giving your answers in the form iab+ where a and b are real numbers. [4] 2 (a) Find the exact roots of the equation 2 7 3 13xx x− +=− . [3] (b) On the same axes, sketch the curves with equations 2 73yx x=−+ and 13 ,yx= − indicating the value of the x-coordinates of any intersections. Hence solve exactly the inequality 2 7 3 13xx x− +<− . [5] 3 The variables x and y are related by the differential equation ( )dπ 3 π tan 0d y yxx +− = . (a) Using the substitution secyz x= , show that d3 d π zz x −= . Hence, show that the particular solution for which 2y= when π 3x= can be expressed in the form cos e a bxyx += where a and b are constants to be determined. [6] (b) For the graph of the solution found in part (a), find the equations of the two vertical asymptotes closest to the y-axis. [2] 4 A curve C has equation 33x y xy A+−= where A is a non-zero constant. (a) Show that any stationary points on C lie on 23yx= . [2] (b) Find, in terms of A, the x-coordinates of the stationary points, and hence determine the range of values of A for which C has two distinct stationary points. [4] (c) Suppose that C has two stationary points. Determ