SJI 2024 Year 6 HL MAA Prelim Examination Paper 3

STUDENT NAME: _____________________________ TEACHER NAME: _____________________________ ST. JOSEPH’S INSTITUTION YEAR 6 PRELIMINARY EXAMINATION 2024 MATHEMATICS: ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 3 Wednesday 21 August 2024 1 hr 1300 – 1400 hrs/ 1630 – 1730 hrs INSTRUCTIONS TO CANDIDATES • Do not open this examination paper until instructed to do so. • Answer all the questions using the foolscap paper provided. • The use of a scientific or examination graphical calculator is permitted in this paper. • TI-Nspire calculators must be in Press-to-Test mode and cleared of all previous data. • TI-84+ graphical calculators must only have permitted apps and be ram cleared. • Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. • A clean copy of the mathematics: analysis and approaches formula booklet is required for this paper. • The maximum mark for this examination paper is [55 marks]. • Number of printed pages = 5. __________________________________________________________________________ FOR MARKER USE ONLY: Q1 Q2 TOTAL /55

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P3 2 Answer all questions in the answer booklet provided. Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. ………………………………………………………………………………………………… 1. [Maximum mark: 24] In this question, you will be exploring the Steiner inellipse, which is a unique ellipse that is tangential to the midpoints of a triangle. Consider the cubic polynomial 32( ) 2p z z z= + − . It is given that 1 2 3, andz z z are the roots of the equation ( ) 0,pz = such that 1 2 3arg arg argz z z−     . In an Argand diagram, points P, Q and R represent the complex numbers 1 2 3, andz z z respectively. (a) (i) Find 1 2 3, andz z z . (ii) Indicate P, Q and R on the Argand diagram. [5] P, Q and R are joined together to form the triangle PQR. It is given that A is the midpoint of PR, B is the midpoint of QR, and C is the midpoint of PQ. (b) Indicate A, B and C clearly on the same Argand diagram. [1] Let 45andzz be the roots of '( ) 0pz = . Points H and K represent 4z and 5z respectively such that AH AK . (c) (i) Solve '( ) 0pz = . (ii) Indicate H and K on the same Argand diagram. [4]

Year 6 Mathematics: