# Chapter 1 - Set Theory

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1 Midterm review Math 3201 Name: Chapter 1 - Set Theory Part 1: Multiple Choice : 1) U = {hockey, basketball, golf, tennis, volleyball, soccer}. If B = {sports that use a ball}, which element would be in B? Basketball Golf Hockey Soccer 2) Which of the following would represent the shaded region? G V G V G V G V 3) Which of the following phrases describes an empty set? Diagram D Common factors of 4 and 12 Prime numbers that are even Multiples of 3 that are less than 12 Factors of 10 that are divisible by 4 4) Set M consists of the multiples of 4 from 1 to 50. Which represents set notation? M= M= M= M= 5) Consider the following sets: R={0, 1, 2, 3, 4, 5, 6} S={2, 4, 6, 8} T={1, 2, 3, 6} Which of the following statements is true? R S R T S R T R 6) Set T is students who like tennis, set B is students who like basketball and set S is students who like swimming, which diagram indicates students who like tennis only? A B C None of these Use the following information to answer #7, #8 & #9 A = {natural numbers from 1 to 10} B = {factors of 12} C = {multiples of 11 less than 100} 7) What is the union of sets A and B,? 8) What is A intersect B, 9) Set B and set C are an example of. sets disjoint empty infinite none of these

2 2 10) What does the 3 represent in this Venn Diagram shown? The number of people who like handball and hockey. The number of people who like handball and hockey but not wrestling. The number of people who like hockey and wrestling and handball. The number of people who like hockey and wrestling but not handball. Part 2Questions: 11. A survey was conducted to determine where people buy coffee. 82 people buy coffee at Tim Hortons. 65 people buy coffee at StarBucks. 17 people did not buy coffee at all. If 130 people were surveyed, how many people buy coffee from ONLY Tim Hortons? 12. There are 36 students who study science. 14 study physics 18 study chemistry 24 study biology 5 study physics and chemistry 8 study physics and biology 10 study biology and chemistry 3 study all three subjects Use a Venn diagram to answer the following questions: (i) (ii) (iii) Determine the number of students who study physics and biology only. Determine the number of students who study at least two subjects. Determine the number of students who study biology only. 13. A grade three teacher asked her class of 28 students about the type of pet they owned. The results were as follows: 28 children have a dog, a cat or a bird 13 children have a dog 13 children have a cat 13 children have a bird 4 children have a dog and cat only 3 children have a dog and bird only 2 children have a cat and bird only A) Algebraically, determine how many children have a dog, cat and bird? B) How many children have only one pet?

3 3 14. In a school of 120 students 5 students took English, Physics, and Chemistry 15 students took Physics and English 8 students took Physics and Chemistry 10 students took English and Chemistry 99 students took English or Chemistry 45 took Chemistry 30 students took Physics James summarized the data using the Venn Diagram shown below: English Physics U Chemistry 19 Identify the regions in James Venn diagram that have errors and describe the errors that James made. Provide a correct Venn diagram with the correct entries. Answers: 1C 2D 3D 4D 5D 6B 7B 8C 9A 10B (i)5 (ii)19 (iii) A)1 B)18 14.Error occurs where subjects overlap Physics Chemistry U English Physics U Biology Chemistry 9 Chapter 2 Counting Methods Part 1. Multiple Choice 1. Samantha is choosing an outfit to wear to the dance. She has 3 different tops, 4 different pants and 2 different pairs of shoes to choose from. How many different outfits could she make from this selection of clothes? A) 9 B) 12 C) 14 D) 24 8! 2. Simplify: 2!5! A) B) 168 C) 336 D) 2,419, Consider the word CAT. In how many different ways can the letters be arranged? A) 1 B) 3 C) 4 D) 6

4 4 ( n 2)! 4. Simplify: n! 1 A) B) 2! C) 2n 3 D) n 2 3n 2 2 n 3n2 5. What restriction must be applied to the variable in #4 above? A) n 2 B) n 2 C) n 0 D) n 0 6. In the grid below, a person must travel from A to B by only heading East (E) or South (S). Under these rules, which represents the total number of possible routes that can be taken to get from A to B? N A) 5! 3!2! B) C) 5! D) 6! 6! 3!2! A B W S E 7. A briefcase lock opens with the correct four-digit code. If the digits 0 to 9 are allowed and a digit cannot be repeated, how many different four-digit codes are possible? A) 24 B) 34 C) 5040 D) How many ways can 5 friends stand in a row for a photograph if Alex and Andrea always stand together? A) 120 B) 48 C) 240 D) How many different arrangements can be made using all the letters in NEWFOUNDLAND? A) 362,880 B) 479,001,600 C) 39,916,800 D) 665, How many different combinations of 3 letters can be made using all 26 letters in the alphabet? A) 15,600 B) 2600 C) 23 D) How many different ways can the letters of SASKATOON be arranged if you must start with a T and end with a K? A) 630 B) 1260 C) 5040 D) The student council has 10 members, 6 girls and 4 boys. A dance committee is to be formed consisting of exactly two girls and two boys. Which calculation could be used to determine the number of different ways this committee could be formed? A) 6 C 2 4 C 2 B) 6 P 2 4 P 2 C) 10 C 4 D) 10 P Solve for n: n 1! 12 n 2! A) 11 B) 13 C) 12 D) Calculate: 2 A) 3 B) 15 C) 30 D) How many different 3 topping pizzas can you create if you have 8 toppings to choose from? A) 56 B) 24 C) 336 D) 6720

5 5 Part 2 Questions: 16. Six different scholarships will be awarded at the graduation ceremonies in November. If the graduating class has 30 students determine the number of ways to award the scholarships if, A) there are no conditions on who can win the scholarships B) no student can win more than one scholarship C) Clara must be awarded the Most Improved Student scholarship and Craig must receive the scholarship for highest average. 17. Four students are to be chosen from a group of 10 to fill the positions of president, vicepresident, treasurer and secretary. In how many ways can this be accomplished? 18. Algebraically solve for n and verify your answer: P n A group of five Art club students are to be selected for a field trip to the Rooms. If there are 5 boys and 6 girls in the club, how many ways can the leader select the five students if there must be at most 2 boys? 20. You need to create a password that must be at least 4 characters but no more than 6 characters long. The password may contain lower case letters as well as the digits 1 to 9. No repeated characters are allowed. How many different passwords are possible? Answers: 1D 2B 3D 4D 5D 6A 7C 8B 9C 10A 11A 12A 13B 14B 15A 16.A) B) C) n= Chapter 3: Probability Part 1 Multiple Choice 1. Given the following probabilities, which event is most likely to occur? P = 0.28 P = P = 0.27 P = 2. Raymond has 12 coins in his pocket, and 9 of these coins are quarters. He reaches into his pocket and pulls out a coin at random. Determine the odds against the coin being a quarter. 1:4 1:3 3:4 3:1 3. Julie draws a card at random from a standard deck of 52 playing cards. Determine the odds in favour of the card being a heart. 3:1 1:3 1:1 3:13 4. Tia notices that yogurt is on sale at a local grocery store. The last eight times that yogurt was on sale it was available only three times. Determine the probability of yogurt being available this time The weather forecaster says that there is a 30% probability of fog tomorrow. Determine the odds against fog. 3:7 3:10 7:3 7:10 6. From a committee of 18 people, 2 of these people are randomly chosen to be president and secretary. Determine the number of ways in which these 2 people can be chosen for president and secretary. 2P 2 2P 1 18P 2 18 P Nine boys and twelve girls have signed up for a trip. Only six students will be selected to go on the trip. Determine the probability that there will be equal numbers of boys and girls on the trip % 22.61% 27.35% 34.06%

6 6 8. Cai tosses four coins. Determine the probability that they all land as tails. 6.25% 12.50% 18.75% 25.00% 9. Select the events that are mutually exclusive. Drawing a red card or drawing a diamond from a standard deck of 52 playing cards. Rolling a sum of 8 or rolling an even number with a pair of six-sided dice, numbered 1 to 6. Drawing a black card or drawing a Queen from a standard deck of 52 playing cards. Drawing a 3 or drawing an even card from a standard deck of 52 playing cards. 10. Helen is about to draw a card at random from a standard deck of 52 playing cards. Determine the probability that she will draw a black card or a spade. 11. Sarah draws a card from a well-shuffled standard deck of 52 playing cards. Then she draws another card from the deck without replacing the first card. Determine the probability that both cards are NOT face cards. 12. Misha draws a card from a well-shuffled standard deck of 52 playing cards. Then he puts the card back in the deck, shuffles again, and draws another card from the deck. Determine the probability that both cards are even numbers. 13. Select the events that are dependent. A. Rolling a 2 and rolling a 5 with a pair of six-sided dice, numbered 1 to 6. B. Drawing an odd card from a standard deck of 52 playing cards, putting it back, and then drawing another odd card. C. Drawing a spade from a standard deck of 52 playing cards and then drawing another spade, without replacing the first card. D. Rolling an even number and rolling an odd number with a pair of six-sided dice, numbered 1 to A five-colour spinner is spun, and a die is rolled. Determine the probability of spinning yellow and rolling a % 3.33% 6.13% 7.75% 15. Two cards are drawn, without being replaced, from a standard deck of 52 playing cards. Determine the probability of drawing a five then drawing a two % 1.227% 1.613% 2.009% Part 2 Questions: 16. A credit card company randomly generates temporary five-digit pass codes for cardholders. Meghan is expecting her credit card to arrive in the mail. Determine, to the nearest hundredth of a percent, the probability that her pass code will consist of five different even digits. 17. From a committee of 18 people, 3 of these people are randomly chosen to be president, vicepresident, and secretary. Determine, to the nearest hundredth of a percent, the probability that Evan, Elise, and Jaime will be chosen. 18. Sonja has letter tiles that spell MICROWAVE. She has selected four of these tiles at random. Determine, to the nearest tenth of a percent, the probability that the tiles she selected are two consonants and two vowels.

7 7 19. Homer hosts a morning radio show in Halifax. To advertise his show, he is holding a contest at a local mall. He spells out NOVA SCOTIA with letter tiles. Then he turns the tiles face down and mixes them up. He asks Marie to arrange the tiles in a row and turn them face up. If the row of tiles spells NOVA SCOTIA, Marie will win a new car. Determine the probability that Marie will win the car. Show your work friends are lining up to get in to see the Hunger Games: Catching Fire movie, including Bob and Sam. A) What is the probability that Bob and Sam will be next to each other. B) What is the probability that they will not be next to each other? 21. The probability that Haley will exercise on Sunday is 0.6. The probability that she will go shopping on Sunday is 0.5. The probability that she will do both is 0.3. Determine the probability that Haley will do at least one of these activities on Sunday 22. The probability that a plane will leave Toronto on time is The probability that a plane will leave Toronto on time and arrive in Saskatoon on time is Determine the probability that a plane will arrive in Saskatoon on time, given that it left Toronto on time. Show your work. 23. Bonita has six identical red marbles and ten identical blue marbles in a paper bag. She pulls out one marble at random and then another marble, without replacing the first marble. Determine the probability that she pulls out a pair of red marbles. 24. A five-colour spinner is spun, and a four-sided die is rolled. Determine the probability of spinning orange and rolling a 1. Answers: 1A 2B 3B 4B 5C 6C 7D 8A 9D 10C 11C 12D 13C 14B 15A % % % % 75% Chapter 4: Rational Expressions and Equations Part 1 Multiple Choice 1. Identify the rational expression that is equivalent to, 2. Determine the non-permissible value(s) for 3. Simplify: 4. Perform the indicated operation and simplify:

8 8 5. Simplify: 6. Find the product: 7. Find the quotient: 8. Simplify: 9. Simplify: Part 2 Questions: 10. Find each product or quotient in simplest form. A) B) C) 11. Find each sum or difference in simplest form. Identify the non-permissible values. A) B) Answers: 1B 2A 3A 4D 5A 6A 7B 8B 9A 10 A) B) C) 5 11A) B)

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