ACSI 2023 Promo Paper 1

FINAL EXAMINATION 2023 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS : ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 1 Friday 22 September 2023 2 hours _________________________________________________________________________ INSTRUCTIONS TO CANDIDATES • Write your session number in the boxes above. • Do not open this examination paper until instructed to do so. • You are not permitted access to any calculator for this paper. • Section A: answer all questions. Answers must be written within the 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 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 [110 marks] This question paper consists of 12 printed pages including this cover page. Section A (53 Marks) Section B (57 Marks) Question Marks Question Marks 1 9 2 3 4 10 5 6 7 11 8 Subtotal Subtotal TOTAL / 110 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 1 Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. 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: 4] Solve the following equation 23𝑥𝑥−2 = 5(3−𝑥𝑥). Give your answer in the form ln 𝑎𝑎 ln 𝑏𝑏, where 𝑎𝑎 and 𝑏𝑏 are integers.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 2 2. [Maximum mark: 4] It is given that 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 3 2, where 𝜋𝜋 2 < 𝑐𝑐 < 3𝜋𝜋 2 . Find the exact value of 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 3 3. [Maximum mark: 7] (a) Show that �(2𝑟𝑟 − 1) 𝑛𝑛 𝑟𝑟=1 = 𝑛𝑛2 [3] (b) Hence, find the value of 11 + 13 + 15 + ⋯ + 99. [4]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 4 4. [Maximum mark: 7] The expansion of (𝑥𝑥 + ℎ)7, where ℎ is a non-zero rational number, can be written as 𝑥𝑥7 + 𝑎𝑎𝑥𝑥6 + 𝑏𝑏𝑥𝑥5 + 𝑐𝑐𝑥𝑥4 + ⋯ + ℎ7 where 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 ∈ 𝑅𝑅. Given that