Discrete Mathematics

DISCRETE MATHEMATICS 1 The matrix below shows the number of crews a construction company uses per building for three types of buildings. Houses Apartments Offices Building Crews 11 45 23 Electrical Crews 3 8 3 Plumbing Crews 4 6 2 Landscaping Crews 1 5 1 The company is currently working on 9 houses, 2 apartment buildings, and 6 office buildings. Which statement is true? A There are more building crews working on offices than on houses. B There are more electrical crews working on apartments than on offices. C There are more plumbing crews working on offices than on apartments. D There are more landscaping crews working on houses than on apartments. 1 Go to the next page. DISCRETE MATHEMATICS 2 The graph below displays a relationship between 7 locations. Q R P S T U V Can an Euler path be drawn for this graph? A no, because there are exactly 2 vertices with an odd degree B no, because each vertex is of an even degree C yes, because there are exactly 2 vertices with an odd degree D yes, because each vertex is of an even degree 2 Go to the next page. DISCRETE MATHEMATICS 3 What is the critical path for the diagram below? C S T A R T 5 E 6 H 3 2 0 A 3 B 7 D F 6 G 1 A START-A-B-C-E-H-J-FINISH B START-A-B-C-E-J-FINISH C START-A-B-D-F-I-J-FINISH D START-A-B-D-G-J-FINISH 3 9 I 2 J 1 8 Go to the next page. F I N I S H DISCRETE MATHEMATICS 4 A student needs to complete the task list below for a project. Task Time Prerequisites Start 0 minutes - 1 5 minutes none 2 10 minutes none 3 10 minutes 1 4 8 minutes 1, 3 5 4 minutes 2, 4 6 10 minutes 4 7 18 minutes 5, 6 8 7 minutes 7 Finish If multiple tasks can be done simultaneously, what is the minimum amount of time it will take the student to complete the task list? A 47 minutes B 52 minutes C 58 minutes D 72 minutes 4 Go to the next page. DISCRETE MATHEMATICS 5 6 Which group and purpose would work best in a census survey? A the students in one class, to determine what students in the school think about school lunches B the workers in an entire office building, to determine the most preferred day for weekly meetings C the citizens of one county, to determine who is preferred nationally in an upcoming presidential election D the citizens of an entire city, to determine what citizens think about tourism in their state The amount of time Mr. Smith spends exercising each day is approximately normally distributed with a mean of 30 minutes and a standard deviation of 10 minutes. On approximately what percent of the days in a year does Mr. Smith exercise for between 10 minutes and 40 minutes? A 50% B 82% C 88% D 95% 5 Go to the next page. DISCRETE MATHEMATICS 7 A stem-and-leaf plot is shown below. Stem Leaves 1 3, 7 2 2, 2, 3, 3, 3, 6, 7, 9 3 1, 2, 4, 8, 8 4 1 5 2 Which best describes the distribution of the data? A skewed right B skewed left C symmetric D bimodal 6 Go to the next page. DISCRETE MATHEMATICS 8 9 A spinner is divided into 12 sections that are each equally likely to occur. The sections are lettered from A to L. Philip will spin the spinner 3 times. What is the probability that the spinner will land on the letter G exactly 1 out of the 3 times? A 121 5,184 B 121 1,728 C 121 576 D 121 288 A school is selecting new members for a council. • There are 10 seniors competing for 6 spots. • There are 8 juniors competing for 5 spots. • There are 9 sophomores competing for 4 spots. • There are 7 freshmen competing for 2 spots. How many unique groupings of new members can the school make to fill the spots? A 413 B 160,986 C 31,116,960 D 2,333,606,220 7 Go to the next page. DISCRETE MATHEMATICS 10 11 A ballot contains a list of 5 candidates. Each voter can choose 0 to 5 candidates. In how many ways can a voter complete the ballot? A 17 B 20 C 32 D 125 Suppose Caleb has an overall probability of 1 of winning at a game of chance 10 each time he plays. What is the approximate probability that Caleb will win the game at least once if he plays it 10 times? A 0.61 B 0.65 C 0.90 D 0.96 8 Go to the next page. DISCRETE MATHEMATICS 12 Thirteen members of the chess club have voted to determine who the new president will be. The table below shows the preference schedule for the four candidates. 6 Votes 5 Votes 2 Votes 1st Place Latesha Maria Kevin 2nd Place Kevin Kevin Maria 3rd Place Maria Latesha Latesha 4th Place Jeff Jeff Jeff The winner of the election will be determined using the plurality method. Kevin had decided to drop out of the election before the votes were counted. What effect, if any, will this have on the results of the election? A There will be no effect on the result of the election. B Maria was in the lead before Kevin dropped out, but Latesha will win once he has dropped out. C Latesha was in the lead before Kevin dropped out, but Maria will win once he has dropped out. D Kevin was in the lead before he dropped out, but Jeff will win once Kevin has dropped out. 9 Go to the next page. DISCRETE MATHEMATICS 13 A company has five board members. The board uses a weighted voting system to make decisions. At least 13 votes are needed to pass a motion. The weight of each board member’s vote is listed below. 7, 5, 4, 2, 1 How many different winning coalitions are there? 14 A 6 B 8 C 9 D 11 A series is shown below. ¥ Σ (2n − 1) n =1 Which is true about the series? A The series converges to –1. B The series converges to 1. C The series converges to 2. D The series diverges. 10 Go to the next page. DISCRETE MATHEMATICS 15 A game uses a spinner to determine if a player gains tokens or loses a token. • The spinner is divided into 10 spaces labeled 1 through 10, each equally likely to occur. • If a player spins and lands on an odd number that is not 9, then the player gains 1 token. • If a player spins and lands on the number 9, then the player gains 2 tokens. • If a player spins and lands on an even number, then the player loses 1 token. What is the expected number of tokens received for each turn? 16 A 0.10 B 0.30 C 1.7 D 1.9 A team plays 8 games in September. For any game, there is a 0.60 probability that the team will win. What is the approximate probability of the team winning 4 to 6 of the games in September? A 0.762 B 0.720 C 0.254 D 0.240 11 Go to the next page. DISCRETE MATHEMATICS 17 Students at a high school are offered three different sports to play, Sport I, Sport II, and Sport III. Which Venn diagram has shading in only the sections where students play exactly two of the sports? A B Sport II Sport I Sport II Sport III Sport I C D Sport II Sport II Sport I Sport III Sport I Sport III 12 Sport III Go to the next page. DISCRETE MATHEMATICS 18 Four students, W, X, Y, and Z, ran for the position of classroom representative. The results of the voting are shown in the preference schedule below. X X W Y W Z W W Y Z X Y Y Z X W Z W Z Y Z X Y X 8 5 4 3 7 6 The winner was chosen using the sequential runoff method. Which student is the winner? A W B X C Y D Z 13 Go to the next page. DISCRETE MATHEMATICS Discrete Mathematics Answer Key Item Number Type2 Key Percent Correct3 Standard 1 MC A 72% 1.01.a 2 MC C 51% 1.02 3 MC C 42% 1.02 4 MC C 35% 1.02 5 MC B 46% 2.01.a 6 MC B 45% 2.01.d 7 MC A 41% 2.01.e 8 MC C 29% 2.02.f 9 MC C 25% 2.02.b 10 MC C 21% 2.02.b 11 MC B 18% 2.02.f 12 MC C 64% 2.03.b 13 MC C 21% 2.03.c 14 MC D 22% 3.01.c DISCRETE MATHEMATICS Item Number Type2 Key Percent Correct3 Standard 15 MC A 30% 2.02.d 16 MC B 40% 2.02.e 17 MC A 84% 2.01.e 18 MC A 50% 2.03.b