MI 9758 2023 Prelim P1

© Millennia Institute 9758/01/PU3/Prelim/23 2023 Preliminary Exams Pre-University 3 MATHEMATICS 9758/01 Paper 1 14 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 question. The total number of marks for this paper is 100. Qn No. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 * Total Score Max Score 4 4 6 6 8 8 8 9 10 12 13 12 100 CANDIDATE NAME ADMISSION NUMBER CLASS

2 © Millennia Institute 9758/01/PU3/Prelim/23 1 A function f is defined as ( ) 3f x ax bx c= ++ . The graph of ( )fyx= passes through the points ( )1, 10 and ( )2, 12− . Given that ( )f x is divisible by 2x− , find the values of a, b and c. [4] 2 Solve the differential equation 2 4 2 d 3,d yxx x = + given that the point 231, 6   is a stationary point of the solution curve. [4] 3 One of the roots of the equation 32 70x ax x b+ − += , where a and b are real numbers, is 2ix= −+ . (i) Find the values of a and b. [3] (ii) Hence, without using a calculator, find the other roots of the equation. [3] 4 (i) Without using a calculator, solve the inequality 2 14 25 x xx − ≥−− . [4] (ii) Hence solve the inequality 2 14 25 x xx − ≥−− . [2] 5 (i) Sketch, on the same diagram, the graphs of 2 ln cos xy x= and 3cos lny xx= for 3 22 xππ<< , showing clearly the equations of any asymptotes and the coordinates of any axial intercepts. [3] (ii) The region R is bounded by the two curves in part (i). Find the area of R , giving your answer to 2 decimal places. [3]

3 © Millennia Institute 9758/01/PU3/Prelim/23 [Turn over (iii) The region R in part (ii) is rotated 2π radians about the x-axis to form a solid S. Find the volume of S. [2] 6 The function f is defined by f: 21 xx x− + , 1,