MI 9758 2025 Prelim P2

©Millennia Institute 9758/02/PU3/25 2025 Preliminary Examination Pre-University 3 MATHEMATICS 9758/02 Paper 2 August/September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer 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. 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 must 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. This document consists of 8 printed pages. H CANDIDATE NAME ADMIN NUMBER CLASS

2 ©Millennia Institute 9758/02/PU3/25 Section A: Pure Mathematics [40 marks] 1 The diagram shows the curve with equation f( )yx= . The curve passes through the points P ( ), 0p , Q ( ), 0q and R ( )0, r . (a) The curve f( )yx= is transformed onto the curve with equation f (2 1)yx=− . Find the coordinates of the points on the graph of f (2 1)yx=− which correspond to the points P, Q and R on curve f( )yx= . [3] (b) It is given that f ( ) d q p I x x= . Find, in terms of I, the area of the finite region bounded by (i) the curve with equation f ( )yx=− and the x-axis, [1] (ii) the curve with equation 1 f ( 3)2yx=+ and the x-axis. [1] (c) Find the value of 0 f ( ) d q xx , justifying your answer. [1] 2 A curve C has parametric equations 2, 4, x t y t= = + where 0 16t . (i) Find the equation of the tangent to C at the point P with parameter p. [3] (ii) Given that the tangent in part (i) passes through the point (0,1) , find the exact coordinates of the point P. [3] (iii) Sketch the curve C. [2] O y x

3 ©Millennia Institute 9758/02/PU3/25 [Turn over 3 (a) The complex numbers z and w satisfy the simultaneous equations 2 * 6 iz z w+ + = + and i3zw−= . Find z and w, leaving your answers in the form iab+ , where a and b are real numbers. [4] (b) The diagram shows the curve with equation f( )yx= , where f ( )x is given by 32 5x mx nx+ + + , and m and n are real constants. The curve crosses the x-axis at the point P. One of the roots of the equation f ( ) 0x = is 2i+ . Determine the coordinates of the point P. [4] 4 A sequence is defined by 21n n nT T T++ =+ wi