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 point on segment BC such that BFEA is a square and CDEF is a rectangle. ABCD is defined as a golden rectangle if CDEF and DABC are similar, such that  . (e) By letting AB = 1, express ED in terms of  . [1] (f) Hence prove that the exact value for  is . [4] This part investigates how the golden ratio  can be expressed as a trigonometric function of a fixed angle in radians. (g) Starting with sin3 sin 2= , show that 24cos 2cos 1 0− − = . [4] (h) Hence show that the angle is related to the golden ratio  by the relation cos 52  = . [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 6 / 2022 Preliminary Exam / Paper 3 3 2. [Maximum mark: 30] This question explores finding the solutions of second order differential equations. Consider the system of linear differential equations of the form: and 2 ,dy duu u ydx dx= =− + where and yu are functions of .x (a) Show that 2 2 2 0.d y dy ydx dx+−= [3] For the differential equation 2 2 20d y dy ydx dx+−= , it is given that rxye= , where r is a constant, is a solution to this differential equation. (b) Show that 2 2 0.rr+ − = [3] (c) Hence find the two values or r, 12 and ,rr where 12 .rr  [2] (d) Verify that 12 r x r xy Ae Be=+ is also a solution to 2 2 20d y dy ydx dx+−= , where A and B are arbitrary constants. [4] For another differential equation 2 2 20d y dy ydx dx− + = , it is given that k xye= , where k is a constant, is a solution to this differential equation. (e) Deduce the possible value(s) of k. [3] (f) Verify that ( ) kxy A Bx e=+ is also a solution to 2 2 20d y dy ydx dx− + = , where A and B are arbitrary constants. [4] (g) Given 1, 2AB== and the values of 12, ,r r k found earlier, sketch the graph of ( ) 12 , kx r x r x A Bx ey Ae Be += + showing clearly any asymptote(s), intercept(s) and stationary point(s). [4] (h) With the aid of the diagram in (g), find the oblique asymptote of the graph in the form ,y mx c=+ justifying your conclusions. [3] (i) Determine the general solution to 2 24 4 0.d y dy ydx dx− + = [4]