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 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 are three consecutive terms of an arithmetic sequence, find the possible values of ℎ.

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 5 5. [Maximum mark: 9] (a) Use mathematical induction to prove that for any real number 𝑥𝑥, |x|<1, and any positive integer 𝑛𝑛, (1 + 𝑥𝑥)𝑛𝑛 ≥ 1 + 𝑛𝑛𝑥𝑥. [6] (b) Hence, show that 0.9994 > 0.996. [3]

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 6 6. [Maximum mark: 6] The graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) is shown in the following diagram. The curve intersects the 𝑥𝑥- axis at (−2,0) and it has a stationary point at (4,0). The equations of the asymptotes are 𝑥𝑥 = 0 and 𝑦𝑦 = 2(𝑥𝑥 − 6). Sketch the curve 𝑦𝑦 = 1 𝑓𝑓(𝑥𝑥) , indicating clearly the equations of any asymptotes, coordinates of any intercepts with the axes and any stationary point(s).

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 7 7. [Maximum mark: 10] In the following diagram, the points 𝐴𝐴, 𝐵𝐵, 𝐶𝐶 and 𝐷𝐷 are on the circumference of a circle with centre 𝑂𝑂 and radius 𝑟𝑟. 𝐴𝐴𝐶𝐶 is the diameter of the circle. 𝐵𝐵𝐶𝐶 = 𝑟𝑟, 𝐴𝐴𝐷𝐷 = 𝐶𝐶𝐷𝐷 and 𝐴𝐴𝐵𝐵� 𝐶𝐶 = 𝐴𝐴𝐷𝐷�𝐶𝐶 = 90°. (a) Show that cos (𝐵𝐵𝐴𝐴̂𝐷𝐷) = √3−1 2√2 . [5] (b) Hence or otherwise, show that the area of triangle 𝐴𝐴𝐵𝐵𝐷𝐷 = 3+√3 4 𝑟𝑟2. [5] 𝑟𝑟 𝑟𝑟

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 8

ACS (Independent) / Mathematics Department / Mathematics AA HL / Year 5 / 2023 Final Examination / Paper 1 9 8. [Maximum mark: 6] Consider the following functions: 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 𝑎𝑎𝑥𝑥 + 𝑏𝑏 𝑔𝑔(𝑥𝑥) = 𝑎𝑎𝑥𝑥2 + 2𝑏𝑏𝑥𝑥 − 2 , where 𝑎𝑎 and 𝑏𝑏 are positive integers. It is given that the solution of 𝑓𝑓(𝑥𝑥) ≥ 𝑔𝑔(𝑥𝑥) is −5 ≤ 𝑥𝑥 ≤ 1, find the values of 𝑎𝑎 and 𝑏𝑏.