2024 JC2 H2 Math prelim - MI

1 © Millennia Institute 9758/01/PU3/Prelim/24 2024 Preliminary Examination Pre-University 3 MATHEMATICS 9758/01 Paper 1 9 September 2024 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. A nswer 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. T he 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. This document consists of 7 printed pages. Qn No. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 * Total Score Max Score 5 4 7 8 9 9 11 11 11 13 12 100 CANDIDATE NAME ADMISSION NUMBER CLASS

2 © Millennia Institute 9758/01/PU3/Prelim/24 1 (i) A quadratic curve passes through the point ( )1, 4−− and has its turning point at ( )2,5 . Find the equation of the curve. [4] (ii) Given instead that a cubic curve passes through the same point ( )1, 4−− and has the same turning point as stated in part (i). Explain whether it is possible to obtain a unique equation of the curve based on given information. [1] 2 Use the substitution 5xu= to find ( ) 25 sin 5 dxx x∫ . [4] 3 A line l has equation 4eyx= , 0x≥ and curve C has equation 2 2e xyx= , 0x≥ . Both l and C intersect at the points with coordinates (0, 0) and 4(2, 2e ) as shown in the diagram below. (a) The region R is bounded by the line l , curve C, and the lines 1x= and 3x= . Find, correct to 4 significant figures, the area of region R. [3] (b) The region S is bounded by the line l and curve C. Show that the volume V of the solid formed when S is rotated 2π radians about the x -axis i s ( ) 8e6V ABπ= + , where A and B are exact constants to be determined. [4] x y O

3 © Millennia Institute 9758/01/PU3/Prelim/24 [Turn over 4 The equation of a curve is ( ) 2 21 .xy y x++ = (i) Find the equations of the two tangents which are parallel to the y -axis. [4] (ii) The normal to the curve at the point A ( )4, 5− meets the curve again at the point B. Find the coordinates of poin