ACSI 2022 Promo Paper 2

FINAL EXAMINATION 2022 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS : ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 2 FRIDAY 23 SEP 2022 2 hours _________________________________________________________________________ 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. • Section A: answer all questions. Answers must be written within the answer boxes provided. • Section B: answer all questions on the writing paper provided. Fill in your session number on each answer sheet, and attach them to this examination paper using the string 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 [110 marks] • Questions with an asterisk (*) are common to both HL and SL papers This question paper consists of 13 printed pages including this cover page. Section A (55 Marks) Section B (55 Marks) Question Marks Question Marks 1 10 2 3 4 11 5 6 7 12 8 9 Subtotal Subtotal TOTAL / 110 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 2 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. Section A Answer all questions. Answers must be written within the answer boxes provided. Working may be continued below the lines, if necessary. 1.* [Maximum mark: 6] In an infinite geometric sequence, 𝑢1 = 𝑎, 𝑢2 = 𝑎2 − 2. (a) Given that 𝑎 = 9 5, show that the sum to infinity of the sequence exists. [2] (b) Find the least value of 𝑛 such that 𝑆∞ − 𝑆𝑛 < 1 100. [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 3 2.* [Maximum mark: 4] The time 𝑇 hours taken for a machine to cool down to temperature 𝑥℃ after it has been turned off can be modelled by 𝑇 = 5𝑙𝑛 ( 13 𝑥−𝑘), where 𝑇 ≥ 0 and 𝑘 is a positive constant. The machine has an initial temperature of 36℃ when it is turned off. (a) Show that 𝑘 = 23. [1] (b) Sketch the graph of the machine’s temperature for 𝑇 ≥ 0, clearly stating the coordinates of any axial intercept and the equation of any asymptote. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 4 3.* [Maximum mark: 6] On 1st January 2022, Sam invested $600 in an account that pays a nominal annual interest rate of 1.2%, compounded monthly. The amount of money in Sam’s account at the end of each year follows a geometric sequence with common ratio, 𝑟. (a) Find the value of 𝑟, giving your answer correct to 4 significant figures. [2] (b) Sam makes no further deposits or withdrawals from the account. Find the year in which the amount of money in Sam’s account becomes more than $1000. [4]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 5 4.* [Maximum mark: 5] The functions 𝑓 and 𝑔 are defined by 𝑓(𝑥) = 3−2𝑥 𝑥+1 , 𝑥 ∈ 𝑅, 𝑥 ≠ −1 and 𝑔(𝑥) = 𝑥 − 𝑎, where 𝑥 ∈ 𝑅, 𝑎 ∈ 𝑅. (a) Find the range of 𝑓. [2] (b) Given that (𝑓 ∘ 𝑔)(𝑥) has an asymptote at 𝑥 = −4 for all 𝑥 ∈ 𝑅, determine the value of 𝑎. [3]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 6 5. [Maximum mark: 7] Consider the quartic equation 𝑧4 − 3𝑧3 + 𝑎𝑧2 + 𝑏𝑧 + 𝑐 = 0, 𝑧 ∈ 𝐶. Two of the roots of this equation are 2 and 1 + 2𝑖. Find the possible values of 𝑎, 𝑏 and 𝑐 where 𝑎, 𝑏, 𝑐 ∈ 𝑅.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 7 6. [Maximum mark: 7] Find the roots of the equation (𝑤 − 2𝑖)3 = 8𝑖, 𝑤 ∈ 𝐶. Give your answers in Cartesian form.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 8 7. [Maximum mark: 7] The following diagram shows two circles of radius 1 unit, with centres at O and A respectively. The circles intersects at B. Diagram not drawn to scale. (a) Find the value of angle AOB, giving your answers in radians. [1] (b) Find the area of the shaded region. Give your answer correct to 3 significant figures. [6]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 9

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Examination / Paper 2 10 8. [Maximum mark: 5] Consider the graph of the function 𝑓(𝑥) = cos(𝑝𝑥 − 𝑞) , 𝑥 ∈ 𝑅 where 𝑝 and 𝑞 are positive constants. The graph of 𝑓 intersects the x-axis at point A, point B and point C. This is shown in the following diagram. The coordinates of B are (1.1, 0) and the coordinates of C are (1.7, 0). (a) Find the coordinates of A. [1] (b) Find the value of 𝑝. [2] (c) Find the smallest positive value of 𝑞. [2]