ASRJC 9758 2023 Prelim P2

[Turn over ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS JC2 Preliminary Examination Paper 2 (100 marks) 9758/02 3 hours Additional Material(s): List of Formulae (MF26) CANDIDATE NAME CLASS / READ THESE INSTRUCTIONS FIRST Write your name and class in the boxes above. Please write clearly and use capital letters. Write in dark blue or black pen. HB pencil may be used for graphs and diagrams only. Do not use staples, paper clips, glue or correction fluid. Answer all the questions and write your answers in this booklet. Do not tear out any part of this booklet. 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. 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. All work must be handed in at the end of the examination. If you have used any additional paper, please insert them inside this booklet. The number of marks is given in brackets [ ] at the end of each question or part question. Question number Marks 1 2 3 4 5 6 7 8 9 10 11 Total This document consists of 21 printed and 3 blank pages.

2 © ASRJC 2023 y x R Section A: Pure Mathematics [40 marks] 1 Relative to the origin O, two fixed points, A and B, have position vectors a and b respectively. The point P has position vector p given by ( )p a1λλ= +− b where λ is a parameter such that 01 λ<< . (a) Show that for all real values of λ, the point P is collinear with A and B. [2] (b) If angle AOB is 090 , show that the position vector of the foot of perpendicular from O to AB is ( ) 2 22+− + aa ba ab . [3] 2 One of the roots of the equation 432 4 78 0z z az bz− + ++= , where a and b are real, is 3 2i+ . Find the values of a and b. Hence find the other roots of the equation, giving your answers in exact form. [6] 3 (a) Integrate by parts twice to show that 22 1e sin d e (2sin cos )5 xx xx x x c= −+∫ . [4] (b) The diagram below shows the shaded region R that is bounded by part of the curve 2 = e sin x yx π− , the line 4 ( 1)xy π= + and the x – axis. The line 4 ( 1)xy π= + cuts the curve and the x – axis at , 12 π  and , 04 π  respectively. Show that the volume generated when region R is rotated 2π radians about the axisx− is given by ( ) 2eAB C πππ −++ , where A, B and C are constants to be determined. [4]

3 © ASRJC 2023 [Turn over 4 The functions f and g are defined as follows: 3f : e 1 xx − − , 3x≤ 22g: , , 0xax x x −∈< , where 01 a<< . (a) Find f −1 in similar form. [3] (b) Sketch the