Math 140: Final Worksheet Instructions: The following worksheet can count as either a bonus sheet for the final exam or a chance to allow the final exam grade to replace the lowest of the two test grades. Circle One Option: 1. Bonus Points on Final Exam: The worksheet will count as raw bonus points on the final exam, for up to a total of 15 points. Partial credit will be given, and the bonus points will be calculated as: 0.15 (worksheet grade). If this option is selected, the lowest test grade will not be replaced by the final exam grade. 2. Chance for Final to Replace Lowest Test Grade: If you receive at least a 70 percent on this worksheet, then the the lowest test grade will be replaced by the final exam grade if it is better. If this option is selected there will be no bonus applied to the final exam. Instructions: For problem 1-5, prove the statements. Do not assume any specific theorems proved in class about even, odd or rational numbers. You may assume that the sum, difference, and product of integers is an integer. You may assume basic algebra laws such as how to multiple fractions, the distributive law, the associative law, etc. Write all proofs in self-contained paragraph form. Problem 1: Suppose p, q R and p 6= 0. If p is a rational number and pq is a rational number, then q is a rational number. Problem 2: Suppose a, b Z. If a2 b2 is an odd number, then either a is even and b is odd, or b is even and a is odd. Problem 3: There is no largest real number less than 10. Problem 4: Suppose r R. If r is irrational, then 3r + r 2 is irrational. Problem 5: Suppose a Z. If there exists an integer k such that a2 = 4k then a is even Problem 6: Find an Euler trail in the following graph. Number the edges in the order of selection. Problem 7: Perform the following matrix multiplication 1 2 3 2 1 4 3 5 2 1 0 3 3 1 2 2 1 0 3 3 4 2 3 1 4 2 1 2 1 2 Instructions: In problems 8-10 you may leave you answer in the form of addition, subtraction, difference and quotients of factorials and combinations. In problem 10, if you decide to write it as decimal, round to four decimal places. Problem 8: Suppose there is a group of 10 people, including Andy and Brian. Andy and Brian refuse to be on a team together. How many different ways are there to form two teams, a team consisting of 3 people and a team consisting of 5 people (insuring that no person is on both teams, and that Andy and Brian, if on a team, aren't on the same team). Problem 9: How many different ways are there of rearranging the letters in the word \"abstractions\" to form distinct strings (e.g. aattrbscsion and abstractsoin are two such rearrangements) Problem 10: If you draw 5 cards from a standard 52 deck of poker cards, what is the probability that you will get a 4-of-a-kind or 3-of-a-kind (but not a full house). An example of 4-of-a-kind: 5, 5, 5, 5, K. An example if 3-of-a-kind: 2, 2, 7, J, 2. An example of a full-house: J, J, 4, 4, 4