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 a