Mathematical Thinking Exam 2 20 November 2017

Transcription

1 Mathematical Thinking Exam 2 20 November 2017 Name: Instructions: Be sure to read each problem s directions. Write clearly during the exam and fully erase or mark out anything you do not want graded. You may use a scientific or graphing calculator for any calculation or derivation unless otherwise stated so long as you show your work leading up to that point and then state that that is what you are doing. You must show all your work to receive full credit unless otherwise stated. Don t cheat. See below. Prohibited: Class Notes Class Handouts Study Guides and Materials Previous exams or exam solutions The Book Any Electronics (excluding approved calculators) This includes, but is not limited to, all: computers, phones, tablets, ebook readers, headphones, audio players, dictionaries, and translators. Other People s Exams Page: Total Points: Score:

2 MATH 101 Exam 2, Page 2 of 8 20 November 2017 Multiple Choice For this section, you must write the letter corresponding to the correct answer in the blank space provided on the right side of the exam. Markings on the question itself will not be given any credit. 1. (4 points) A path that starts and ends at the same vertex of a graph is a A. Circuit B. Map C. Pie D. Digraph 2. (4 points) A graph where every vertex is connected to every other vertex by exactly one edge is called a: A. Digraph B. Path C. Tree D. Complete Graph 3. (4 points) The Critical Path of a digraph is the path of A. longest length B. shortest length C. most vertices D. most edges 4. (4 points) The problem of a snow plow discerning an efficient way to plow the town streets after a big snow storm is equivalent to the A. Traveling Salesman Problem B. Chinese Postman Problem C. The Greedy Algorithm Problem D. The Critical Path Problem 5. (4 points) If a graph has 4 vertices with odd valence, what is the minimum number of edges that could be needed to Eulerize the graph? A. 0 B. 2 C. 3 D (4 points) The Edge-Walker technique is a process to do what to a graph? A. Connect B. Disconnect C. Eulerize D. Hamiltonize 7. (4 points) The exterior angle of a regular decagon has measure: A. 10 B. 30 C. 36 D out of a possible 28 points

3 MATH 101 Exam 2, Page 3 of 8 20 November (4 points) A quadrilateral with two pairs of parallel sides, all sides of equal length, but possibly having angles with different measures is called what? A. Rectangle B. Triangle C. Trapezoid D. Rhombus 9. (4 points) How many regular tilings are there? A. One B. Two C. Three D. Infinitely many 10. (4 points (bonus)) Dr. Kahn s favorite holiday is Thanksgiving. Mark a in the answer line for extra credit. Gobble gobble and have a great holiday. A. Turkey B. Stuffing C. Potatoes D. Apple Pie Computations 11. Shelby owns 5 shirts, 4 pairs of pants, and 9 hats. (a) (5 points) How many different outfits of 1 shirt, 1 pair of pants, and 1 hat can Shelby wear on a given day? (b) (5 points) Assuming 3 shirts and 1 pairs of pants are dirty and so Shelby can t wear them, how many outfits can she wear? (Assume all hats are clean.) out of a possible 18 points

4 MATH 101 Exam 2, Page 4 of 8 20 November 2017 Diagrams 12. (8 points) Is it possible to have a vertex type If yes, then draw one and if not, then explain why not using full sentences and worked out calculations. 13. (8 points) Is it possible to create a semi-regular tiling of the plane using regular octagons and squares? If yes, then create one showing at least two full rows and columns. If not, explain why not using full sentences. out of a possible 16 points

5 MATH 101 Exam 2, Page 5 of 8 20 November Draw a connected graph with 8 vertices that satisfies each of the requested properties. (a) (4 points) Each vertex has valence 4. (b) (4 points) The valences are: 1,1,2,2,3,3,4,6. (c) (4 points) the graph has an Hamiltonian circuit but no Euler circuit. out of a possible 12 points

6 MATH 101 Exam 2, Page 6 of 8 20 November Consider the below graph for this problem. G K 289 J 425 H (a) (5 points) Using nearest neighbor, solve the TSP on the graph below starting at vertex G. I just want to see the Hamiltonian circuit, do not calculate cost. (b) (5 points) Using sorted edges, solve the TSP on the graph below. I just want to see the work for the sorted edges algorithm and the Hamiltonian circuit, do not calculate cost. out of a possible 10 points

7 MATH 101 Exam 2, Page 7 of 8 20 November (8 points) Determine a minimum cost spanning tree of the graph below using Kruskal s algorithm. Do not calculate the cost, I just want to see the work for Kruskal s algorithm, the redrawn vertices and the spanning tree connecting them. A 160 F B E 311 C 521 D out of a possible 8 points

8 MATH 101 Exam 2, Page 8 of 8 20 November 2017 Short Response 17. (8 points) Construct a digraph with six tasks that has exactly two critical paths of length 20. Then discuss a situation where your digraph could reasonably represent the time to complete a series of tasks to complete a single project. In other words, give the task list with believable times. out of a possible 8 points