YIJC 9758 2023 Prelim P2

This document consists of 22 printed pages and 2 blank pages. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 2 Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/02 13 SEPTEMBER 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 9758/02/JC2PE/23 Section A: Pure Mathematics [40 marks] 1 The diagram shows a sketch of the curve f( )yx= . The shaded region under the curve between 1x = and 4x = , shown in the diagram, is A . This region is split into 4 vertical strips of equal width, h units. (a) State the value of h and show, by drawing on the above diagram , that [ ] 4 1 f (1 ) n nh h = +∑ is less than the area of A. [2] (b) Find a similar expression that is greater than the area of A. [1] You are now given that f ( ) 2 ln( 1) 7xx = − ++ . (c) Use the expression in part (a) and your expression in part (b) to find lower and upper bounds for the area of A. [2] (d) The curve with equation g( )yx= is obtained when the curve with equation f( )yx= is translated by b units in the positive y-direction, where b is a positive constant. Given that [ ] [ ] 44 11 g(1 ) f (1 ) nn nh h nh h c = = += ++∑∑ , where c is a positive constant, express c in terms of b. [1] 2 The complex number z is given by 3i 3 iz k −= − , where k is a real constant. (a) Given that 1k = , find the exact values of z and arg( )z . [4] (b) Given instead that zz ∗= , find the value of k. [3] y x 1 4 O

3 ©YIJC 9758/02/JC2PE/23 [Turn Over 3 (a) Show that 2 1ln 1 r  −