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] Consider Γ (1 2). (f) By using the substitution 𝑡 = 1 2 𝑦2, express Γ (1 2) as an integral in terms of 𝑦. [5] The probability density function of the random variable 𝑍 ∼ N(0,1) is given by 𝑓(𝑧) = 1 √2𝜋 e−1 2𝑧2 . (g) Deduce, with justification, the value of [2] (h) Hence, show that Γ (1 2) = √𝜋. [2] (i) Hence, find the exact value of Γ (3 2). [2] ∫ 𝑓(𝑧)d𝑧 ∞ 0 .

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P3 3 2. [Maximum mark: 30] In this question you will investigate the geometrical properties of complex numbers and roots of unity. Consider 𝑧 = 1 + √2 + i, where i2 = −1. (a) Find the exact value of 𝑧2, leaving your answer in the form 𝑥 + i𝑦, where 𝑥, 𝑦 ∈ ℝ. [2] (b) Find the exact value of arg 𝑧2. [2] (c) Find the exact value of arg 𝑧. [2] (d) Hence, find the exact value of tan 𝜋 8. [2] (e) For any two complex numbers 𝑧1 = 𝑥1 + i𝑦1 and 𝑧2 = 𝑥2 + i𝑦2, (i) show that |𝑧1 − 𝑧2| = √(𝑥1 − 𝑥2)2 + (𝑦1 − 𝑦2)2. (ii) Hence, interpret |𝑧1 − 𝑧2| geometrically. [2] Let 𝜔 = 𝑧 |𝑧|. (f) Show that |𝜔𝑘| = 1, for any integer 𝑘. [1] (g) Sketch the 16 points representing the complex numbers 𝜔𝑘, for 𝑘 = 0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, 8, in an Argand diagram. [2] (h) The points in the Argand diagram represented by the complex numbers 𝜔−5, 𝜔−1, 𝜔3 and 𝜔7 form a square. Find the area of this square. [2] (i) Find the exact value of |𝜔 − 1 𝜔| and of |𝜔2 − 1 𝜔2|. [5] (j) Show that |𝜔 − 1 𝜔| |𝜔3 − 1 𝜔3| = √2. [4] (This question continues on the following page)

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2022/P3 4 (Question 2 continued) Define ∏ 𝑎𝑘 𝑛 𝑘=1 = 𝑎1 × 𝑎2 × ⋯ × 𝑎𝑛. (k) Copy and complete the table below: [6] 𝑛 4 7 10 ∏ |𝜔𝑘 − 1 𝜔𝑘| 𝑛 𝑘=1 = End of Paper