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TE2003/TEE2003 NATIONAL UNIVERSITY OF SINGAPORE T E2003/ TEE 2003 – Advanced Mathematics for Engineers (Semester 1: AY 2019/2020) Time Allowed: 2 Hours INSTRUCTIONS TO STUDENTS 1. Please write your Student Number only. Do not write your name. 2. This assessment paper contains FOUR (4) questions and comprises T H R E E (3) printed pages. 3. All questions carry equal marks. 4. Students are required to answer ALL questions. 5. Students are to write the answers for each question on a new page. 6. This is a CLOSED BOOK assessment. 7. One single sheet of A4 size notes is allowed. 8. Non-programmable calculators are allowed. 2 TE2003/TEE2003 Question 1 There are two fair dice: one is 6-sided, and the other is 4-sided. A person randomly selects one dice and then rolls the selected dice twice. (a) Calculate the probability that the sum of the two dice roll outcomes is equal to 2. (5 Marks) (b) Let A be the event that the maximum of the two dice roll outcomes is 4, and B be the event that the 6-sided dice is selected. Calculate the probability P(B|A). (10 Marks) (c) Let X be the event that the 4-sided dice is selected, and Y be the event that the outcomes of the two dice rolls are both even. Are X and Y statistically independent? (10 Marks) Question 2 There are four boxes numbered A, B, C, and D. Five balls, numbered from 1 to 5, are to be placed in the four boxes at random. Each box can contain any number of balls. (a) Calculate the probability that one box contains all the five balls. (5 Marks) (b) Calculate the probability that ball 1 and 2 are in box A. (5 Marks) (c) Calculate the probability that box A contains no balls. (5 Marks) (d) Calculate the probability that box A contains two balls, while each of the other three boxes contains one ball. (5 Marks) (e) Let X be the event that box A contains ball 1 and 2, and Y be the event that box B contains two balls. Calculate the probability P(Y|X). (5 Marks) 3 TE2003/TEE2003 Question 3 Let 𝑋𝑋~Exp(πœ†πœ†0 ) with πœ†πœ†0 > 0. (a) Calculate the probability P(1 < 𝑋𝑋 < 10). (5 Marks) (b) Calculate the probability P(𝑋𝑋 > 10 | 𝑋𝑋 > 5). (5 Marks) (c) Let 𝑍𝑍 = 𝑋𝑋, if 𝑋𝑋 < 1, and 𝑍𝑍 = 0, if 𝑋𝑋 β‰₯ 1. Calculate the mean of 𝑍𝑍. (7 Marks) (d) Let π‘Œπ‘Œ = 𝑋𝑋 2 . Find the PDF of π‘Œπ‘Œ. (8 Marks) Question 4 (a) Let 𝐢𝐢: 𝑧𝑧(𝑑𝑑) = 𝑑𝑑 + 𝑖𝑖, 0 ≀ 𝑑𝑑 ≀ 1, and 𝑓𝑓(𝑧𝑧) = 𝑧𝑧. Calculate the value of the complex integral, ∫𝐢𝐢 𝑓𝑓(𝑧𝑧)𝑑𝑑𝑑𝑑. (7 Marks) (b) Calculate the value of the complex integral, 𝑧𝑧 2 + 1 𝑑𝑑𝑑𝑑. 2 βˆ’ 6𝑧𝑧 + 8 𝑧𝑧 |𝑧𝑧|=3 οΏ½ (8 Marks) (c) Calculate the value of the complex integral, 𝑧𝑧sin𝑧𝑧 𝑑𝑑𝑑𝑑. 3 |𝑧𝑧|=2 (𝑧𝑧 + 1) οΏ½ – END OF PAPER – (10 Marks) ...
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