YIJC 9758 2023 Prelim P1

This document consists of 24 printed pages. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 1 Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/01 30 AUGUST 2023 3 hours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name 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. 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. 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 mathem atical 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. For Examiners’ Use Question Marks 1 2 3 4 5 6 7 8 9 10 11 Presentation Total / 100

2 ©YIJC 9758/01/JC2PE/23 1 A function is defined as f ( ) ln , 3 ax xa =  + where 1.a > Describe fully a sequence of transformations which transforms the curve lnyx= onto the curve f ( ).yx= [4] 2 Do not use a calculator in answering this question. One of the roots of the equation 32 ,3 13 0z z kz− ++= where k is real, is 2 3i.+ (a) Find the other roots of the equation and the value of k. [4] (b) Deduce the roots of the equation 32i 3 i 13 0.w w kw− + + += [2] 3 (a) Find the exact roots of the equation 23 8 33 .xx x+ −= − [3] (b) On the same axes, sketch the curves with equations 23 83yx x= +− and 3.yx= − Hence solve exactly the inequality 23 8 33 .xx x+ − ≥− [4] 4 A curve C has parametric equations 23 sin 2 , 4cos ,x ty t= = for 0. t π≤≤ (a) Show that d 3 tan 2 ,d y ktx = where k is a constant to be determined. [2] (b) Find the equations of the tangents to C at the points where and .43tt ππ= = [4] (c) Find the acute angle between these two tangents. [2] 5 (a) An infinite geometric progression has first term a and common ratio r , where a and r are non-zero. The sum of all the terms after the nth term of the progression is equal to twice the nth term. Show that the sum to infinity of the progression is three times the first term. [3] (b) The positive integers, starting at 1, a