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2021 Mathematics Final Exam With Answers (40 Solved Questions)

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    1 1 1/34 Module 8: Final Exam Due No due date Points 200 Questions 40 Time Limit 400 Minutes Attempt History Attempt Time Score LATEST Attempt 1 111 minutes 155 out of 200  Correct answers are hidden. Score for this quiz: 155 out of 200 Submitted Nov 1 at 9:56pm This attempt took 111 minutes. 5 / 5 pts Question 1

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    1 1 2/34 Consider the following graph: (Section 4.4) Alt text: A graph with 4 vertices and 7 edges connecting them. What is the minimum number of edges that must be added in order for the graph to contain an Euler path? 1 2 3 4

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    1 1 3/34 Correct. Note that: Adding just one edge that connects two vertices each with odd degree, will leave the graph with only two vertices of odd degree. Hence, the new graph will contain an Euler path. 5 / 5 pts Question 2 A high school math department offers a total of 5 classes: MTH01, MTH02, MTH03, MTH04, and MTH05. However, the following courses cannot be taught at the same time due to staffing limits: • MTH01 conflicts with MTH02, MTH05 • MTH02 conflicts with MTH01, MTH03 • MTH03 conflicts with MTH02, MTH04 • MTH04 conflicts with MTH03, MTH05 • MTH05 conflicts with MTH01, MTH04 What graph could we use to model this situation? A graph with 5 vertices that represent the classes which have a conflict and edges connecting all classes. A graph with 5 vertices that represent the classes and edges connecting classes which have a conflict. A graph with 5 vertices that represent the classes and edges connecting all classes. A graph with 5 vertices that represent the classes and edges connecting two of the classes.

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    1 1 4/34 Correct. Note that: We could create a graph with 5 vertices that represent the classes, and edges connecting classes which have a conflict. 5 / 5 pts Question 3 Which of the following would be the correct way to begin a proof by contraposition of the statement, “If there is snow, then it is winter”? Suppose it is winter. Suppose there is snow. Suppose it is not winter. Suppose there is no snow. Correct. Note that: In a proof by contraposition, we begin by assuming the negation of the conclusion. In this case, this would be "it is not winter." 5 / 5 pts Question 4 Consider the statement, “For all integers a and b , if the product of a and b is even, then a is even or b is even.” What would be the beginning of a proof by contradiction of this statement? Assume that the product of a and b is odd, and a and b are both odd.

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    1 1 5/34 Assume that the product of a and b is even, and a and b are both even. Assume that the product of a and b is even, and a and b are both odd. Assume that the product of a and b is odd, and a is even or b is odd. Correct. Note that: We must assume the hypothesis and the negation of the conclusion. Hence, we begin as follows: Assume that the product of a and b is even, and a and b are both odd. 5 / 5 pts Question 5 Consider the statement, “For all integers a and b , if the product of a and b is even, then a is even or b is even.” What would be the beginning of a proof by contrapositive of this statement? Assume that either a or b are odd. We must show that ab is odd. Assume that a and b are both odd. We must show that ab is even. Assume that a and b are both even. We must show that ab is even. Assume that a and b are both odd. We must show that ab is odd. Correct. Note that: A proof by contraposition would begin as follows: Assume that a and b are both odd. We must show that ab is odd. 0 / 5 pts Question 6 Incorrect Incorrect

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    1 1 6/34 Which of the following is logically equivalent to the statement, “If a book is returned late, then there is a fee?” If there is no fee, then the book is returned late. If there is no fee, then the book is not returned late. If there is a fee, then the book is not returned late. If there is a fee, then the book is returned late. Actually, "If there is no fee, then the book is not returned late" is the contrapositive of the statement; so, we know it is logically equivalent. 0 / 5 pts Question 7 Incorrect Incorrect A high school math department offers a total of 5 classes: MTH01, MTH02, MTH03, MTH04, and MTH05. However, the following courses cannot be taught at the same time due to staffing limits: • MTH01 conflicts with MTH02, MTH05 • MTH02 conflicts with MTH01, MTH03 • MTH03 conflicts with MTH02, MTH04 • MTH04 conflicts with MTH03, MTH05 • MTH05 conflicts with MTH01, MTH04 We want to know the minimum number of time slots needed to teach these classes. What information about the graph would answer this question? number of vertices number of edges

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    1 1 7/34 chromatic index chromatic number Actually, the chromatic number of this graph would tell us how many time slots are needed to teach these classes. 5 / 5 pts Question 8 Consider the statement, “For all integers a and b , if the product of a and b is even, then a is even or b is even.” What would be the beginning of a direct proof of this statement? Assume that a is even or b is even. Then a or b = 2 k for some integer k. We must show that either a is even or b is even, so it suffices to show one or the other. Assume that ab is even. Then ab = 2 k +1 for some integer k. We must show that either a is even or b is even, so it suffices to show one or the other. Assume that ab is even. Then ab = 2 k for some integer k. We must show that either a is even or b is even, so it suffices to show one or the other. Assume that ab is even. Then ab = 2 k for some integer k. We must show that a is even and b is even.

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    1 1 8/34 Correct. Note that: A direct proof would begin as follows: Assume that ab is even. Then ab = 2 k for some integer k. We must show that either a is even or b is even, so it suffices to show one or the other. 0 / 5 pts Question 9 Incorrect Incorrect A high school math department offers a total of 5 classes: MTH01, MTH02, MTH03, MTH04, and MTH05. However, the following courses cannot be taught at the same time due to staffing limits: • MTH01 conflicts with MTH02, MTH05 • MTH02 conflicts with MTH01, MTH03 • MTH03 conflicts with MTH02, MTH04 • MTH04 conflicts with MTH03, MTH05 • MTH05 conflicts with MTH01, MTH04 What is the minimum number of time slots needed to teach these classes? 3 4 5 6 Actually, by trial and error, we see that the circuit can be colored with 3 colors. Hence, a minimum of 3 time slots are required. 5 / 5 pts Question 10

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    1 1 9/34 Which of the following graphs has exactly 5 edges? (Check all that apply.) (Section 4.1) Alt text: A graph with five vertices that are all connected to each other by edges. (Section 4 1)

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    1 1 10/34 (Section 4.1) Alt text: A graph with 6 vertices and 5 edges connecting them to form a path. *c. (Section 4.1) Alt text: A graph with five connected vertices that form a pentagon. d.

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