SJI 2022 Year 6 HL MAA Prelim Examination Paper 3

STUDENT NAME: _____________________________ TEACHER NAME: _____________________________ ST. JOSEPH’S INSTITUTION YEAR 6 PRELIMINARY EXAMINATION 2022 MATHEMATICS: ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 3 Monday 22 August 2022 1 hour 0800 – 0900 hrs INSTRUCTIONS TO CANDIDATES • Write your session number in the boxes above. • Write your name and teacher’s name in the spaces provided. • Do not open this examination paper until instructed to do so. • Answer all the questions using the foolscap paper provided. • The use of a scientific or examination graphical calculator is permitted in this paper. • TI-Nspire calculators must be in Press-to-Test mode and cleared of all previous data. • TI-84+ graphical calculators must only have permitted apps and be ram cleared. • 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 [55 marks]. • This question paper consists of 4 printed pages including the cover sheet. ___________________________________________________________________ FOR MARKER USE ONLY: Q1 Q2 TOTAL /55

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P3 2 Answer all questions in the answer booklet provided. Please start each question on a new page. 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. ………………………………………………………………………………………………… 1. [Maximum mark: 25] In this question you will investigate the Gamma function and some of its properties using calculus and methods of proofs. The Gamma function, Γ(𝑛), is defined as Γ(𝑛) = ∫ 𝑡𝑛−1e−𝑡d𝑡 ∞ 0 , for any number 𝑛 ∈ ℝ except non-positive integers. (E.g., 𝑛 ≠ 0, −1, −2, … ) To approximate Γ(𝑛) using technology for some value of 𝑛, replace “∞” by “999999”. (a) Find the value of Γ(1), of Γ(2), of Γ(3), of Γ(4), of Γ(5) and of Γ(6). [3] (b) Using a factorial, formulate a conjecture on the value of Γ(𝑛) for any positive integer 𝒏. [1] Consider 𝑓(𝑥) = 𝑥𝑛e−𝑥, for any 𝒏 > 𝟎. (c) By sketching the graph of 𝑦 = 𝑓(𝑥), deduce what happens to 𝑓(𝑥) as 𝑥 → ∞. [2] (d) Using integration by parts, show that Γ(𝑛 + 1) = 𝑛 Γ(𝑛), for any 𝒏 > 𝟎. [3] (e) Hence, using mathematical induction, prove your conjecture in (b). [5] Con