MI 9758 2023 Prelim P2

© Millennia Institute 9758/02/PU3/Prelim/23 2023 Preliminary Examination Pre-University 3 MATHEMATICS 9758/02 Paper 2 19 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your admission number, name and class 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. Give 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 questio n. The total number of marks for this paper is 100. This document consists of 28 printed pages. Qn No. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 * Total Score Max Score 5 8 8 8 11 4 7 7 7 9 13 13 100 CANDIDATE NAME CLASS ADMISSION NUMBER

2 © Millennia Institute 9758/02/PU3/Prelim/23 Section A: Pure Mathematics [40 marks] 1 (i) It is given that 3d 2d yx x xy yx =++ , 0x≠ . Using the substitution 2y vx= , show that the differential equation can be reduced to d 1d v vx = + . [2] (ii) Given that 1y= when 1x= , solve the differential equation 3d 2d yx x xy yx =++ . [3] 2 (a) Given that ( ) 2f secxx= , find ( )f x′ and ( )f x′′ . Hence, find the Maclaurin series for ( )f x , up to and including the term in 2x . [5] (b) Using standard series from the List of Formulae (MF26), expand ( ) 4 1 ax − + as far as the term in 2x , where a is a non- zero constant. Hence find the value of a for which the coefficients of the x and 2x terms in the expansion are equal. [3] 3 (a) A sequence 234, , ,...uuu is such that 1ln 1 n nu n −=  + where 2n≥ . Find in terms of N , an expression for NS , where 234 1 ...N NNS uuu u u −=++ + + , leaving your answer in the form ( )ln faN− , where a is a positive constant. [3] (b) (i) Find the set of values of x for which the series 1 3 1 n n x x ∞ =   +∑ has a finite sum. [3] (ii) Given that 0.25x= , find the value of 1 3 1 n n x x ∞ =   +∑ . [2] 4 The curve C has equation 222 20xy+= . (i) Sketch curve C, giving the exact coordinates of any points where the curve meets the axes.