ACSI 2019 Promo Paper 1

FINAL EXAMINATION 2019 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 1 24 September 2019 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 in 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 must be given exactly or correct to three significant figures. • A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required for this paper. • The maximum mark for this examination paper is [100 marks] This question paper consists of 10 printed pages including this cover page. Section A (50 Marks) Section B (50 Marks) Question Marks Question Marks 1 8 2 3 9 4 5 6 10 7 Subtotal Subtotal TOTAL / 100 Candidate Session Number 0 0 2 3 2 9

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / 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 in the boxes provided. Working may be continued below the lines if necessary. 1. [Maximum mark: 5] Find the value of k if � 𝑘𝑘 2𝑟𝑟 = 5 ∞ 𝑟𝑟=0 . .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 2 2. [Maximum mark: 5] Consider the equation (ℎ − 1)𝑥𝑥2 − 𝑥𝑥ℎ + (ℎ + 1) = 0 , where ℎ is a real constant. Find the set of values of h for which this equation has real roots.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 3 3. [Maximum mark: 8] The sum of the first n terms of a sequence {un} is given by 𝑆𝑆𝑛𝑛 = 𝑛𝑛2 + 𝑟𝑟2𝑛𝑛 − 3 4 𝑛𝑛, where 𝑛𝑛 ∈ 𝑍𝑍+ and 𝑟𝑟 ∈ 𝑅𝑅 (a) Write down the value of 𝑢𝑢1in terms of r. [1] (b) Prove that {un} is an arithmetic sequence, stating clearly its common difference. [7]

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2019 Final Exam / Paper 1 4 4. [Maximum mark: 8] Use Mathematical Induction to prove that (1 + 𝑥𝑥)𝑛𝑛 ≥ 1 + 𝑛𝑛𝑥𝑥 for {𝑛𝑛: 𝑛𝑛 ∈ 𝑍𝑍+} where 𝑥𝑥 > 1.

ACS (Independent) / Mathematics Department / Mathematics HL / Yea