DHS 9758 2023 Prelim P1

© DHS 2023 This document consists of 19 printed pages and 5 blank pages. [Turn over Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2023 Year 6 MATHEMATICS 9758/01 Paper 1 13 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number, name and class on 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. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question spe cifically 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. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total Score Max Score 4 6 6 7 9 9 10 11 12 12 14 100

2 DHS 2023 Year 6 H2 Mathematics Preliminary Examination 1 For a > 4, show that 2 32 0 12 3 2 d (4 9 12 56) a xx x aa a k−− = − − +∫ where k is an integer to be found. [4] 2 The curve C has equation 2 33 .1 xxy x −+= − (a) Sketch C, stating the coordinates of any turning points and points of intersection with the axes, and the equations of any asymptotes. [3] (b) Determine the set of exact values of k such that C and the line y kx= has two points of intersection for all real values of x. [3] 3 The equation 3 2 0,z zk− += where k is a real constant, has a root 1 i,za= + where a is a positive real constant. (a) Deduce another root in terms of a. [1] (b) Find the values of a and k, showing your working. [4] (c) Hence find the area of the triangle formed by all the roots on an Argand diagram. [1] 4 (a) Find the set of values of α for which the expression 2 isin2 1 2isin2 α α − + is purely real. [3] (b) If z and w are different non-zero complex numbers and 1,w = find ( ) * 3 .1 wz zw − − [4] 5 With respect to an origin O, the points A and B have position vectors a and b respectively where a and b are non-parallel. It is given that B lies on the line segment AC such that 3 .BC µ  → = −ba (a) Find the value of .µ Hence find OC  → in terms of a and .b [3] (b) Q is a point on line segment OC where .OQ t OC  → 