ACSI 2022 HL AA Prelim Paper 3 Questions

PRELIMINARY EXAMINATION 2022 YEAR 6 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 3 Thursday, 15 September 2022 1 hour _________________________________________________________________________ INSTRUCTIONS TO CANDIDATES • Write your session number in the boxes above. • Do not open this examination paper until instructed to do so. • A graphic display calculator is required for this paper. • Answer all the questions in the answer booklet provided. • Unless otherwise stated in the question, all numerical answers must 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] This question paper consists of 4 printed pages including this cover page. Question Marks 1 2 Total / 55 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2022 Preliminary Exam / Paper 3 1 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: 25] This question explores some interesting relationships between the Golden Ratio  , the arguments of complex roots on an Argand diagram and trigonometric functions. The Golden Ratio  is a very special number. It can be seen almost everywhere, from nature to the human body. It has a numerical value (3sf) approximately. Consider the equation 10 1z = , where .z (a) Show that 5i e  = is a root. [2] (b) Given that every integer can be expressed as , where and show that n is also a root of and hence explain clearly why there are 10 distinct roots. [3] (c) For , mark the coordinates of the roots of 10 1z = on a unit circle in an Argand Diagram. Label each root as . Explain why  , the angle between adjacent pair of roots, is equal to . [3] (d) Using your Argand diagram, show that sin 4 sin= , cos4 cos=− , sin3 sin 2= and hence tan3 tan 4 tan tan 2   = . [5]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2022 Preliminary Exam / Paper 3 2 This part investigates the derivation of the exact value of the golden ratio  . Consider the rectangle ABCD, with width AB length BC. E is a point on segment AD and F is a po