ACJC 9758 2023 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/01 Paper 1 21 August 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name 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. Write 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. The use of an appr oved graphing calculator is expected, where appropriate. 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. This document consists of 27 printed pages and 1 blank page. Question Marks 1 /3 2 /4 3 /5 4 /6 5 /7 6 /8 7 /9 8 /10 9 /11 10 /12 11 /12 12 /13 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 [Turn over

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 [Turn Over 1 A curve has equation ( )fyx= , where ( ) 32f x x ax bx c=+ ++ . The curve ( )fyx= has a maximum point at ( )1, 32− , while the curve ( ) 1 fy x= has a vertical asymptote with equation 5x= . Find the equation of the curve ( )fyx= . [3] 2 Do not use a calculator in answering this question. The complex numbers z and w satisfy the following equations. * 2 4i 2 3i wz zw + = −+ += Find z and w, giving your answers in the form iab+ , where a and b are real numbers. [4] 3 Let ( )f 1 xx x = + . (i) Show that ( ) ( ) 1 0 2f d 22 3xx = −∫ . [2] (ii) The diagram below shows a sketch of the curve with equation ( )fyx= . Rectangles width 1 n are drawn as each of shown. By considering the total area of these n rectangles, deduce that [3] y x 0 ……… 1

4 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 4 The points P, Q and R have position vectors p, q and r respectively, with respect to the origin O. The vector p is a unit vector and the vectors q and r are non-zero vectors. (i) Write down the geometrical meaning of ⋅pq . [1] It is given that 2⋅=pq and that the point M is the foot of the perpendicular from Q to the lin