MTH 201 Applied Mathematics Sample Final Exam Questions 1. The augmented matrix of a system of equations (in two variables) is: 2 1 6 4 2 12 Which of the following is true about the system of equations? (a) The system of equations has a unique solution. (b) The system of equations has infinite solutions. (c) The system of equations has no solutions. (d) Not enough information to answer the question 2. Which of the following is the augmented matrix of the system of equations? 2 7 = + =0 =2 2 1 7 (a) 1 1 0 1 0 1 2 2 1 7 (b) 1 1 0 1 1 2 0 2 1 1 (c) 1 1 0 7 1 1 2 2 1 1 (d) 1 1 0 7 0 1 2 1 4 2. Let = 2 5 5 1 Findthematrixafterthesequence of row transformations (2) 1 + 0 0 1 2 2 2 and ( ) + 2 2 on (a) 1 4 2 0 1 0 1 0 0 1 2 (b) 1 4 2 0 1 0 1 0 0 1 2 (c) 1 4 2 0 1 8 5 0 0 1 2 1
(d) 1 4 2 0 1 0 5 0 0 1 2 4. Let = 4 2 2 1 4 and = 0 5 1 8 2 What is the (1 ) entry of 2? 2 0 5 0 2 7 (a) 7 (b) 17 (c) 12 (d) 9 5. Let = 1 2 2 0 2 6 and = 1 5 0 5 What is the (2 ) entry of? 2 0 8 1 1 0 (a) 6 (b) 0 (c) 24 (d) (2 ) entry of does not exist (e) None of the above 6. The reduced form of the augmented matrix of a system of equations in the variables and is 1 0 1 0 1 2 0 Which of the following is a solution of the system of equations? 0 0 0 0 (a) ( 2 ) where is any real number. (b) ( + 2 ) where is any real number. (c) ( 2 ) where is any real number. (d) The system of equations has no solution. (e) none of these 7. The reduced form of the augmented matrix of a system of equations in the variables and is 1 1 0 2 0 0 1 4 Which of the following is true about the system of equations? 0 0 0 0 (a) A solution of this system of equation is of the form (2 + 2 4 ) where is any real number (b) A solution of this system of equation is of the form (2+ 4) where is any real number (c) A solution of this system of equation is of the form (2 + 2 4 ) where is any real number (d) The system of equations has no solution. (e) None of the above 2
8. The graph of an inequality is Find the inequality. (a) + (b) (c) (d) + 9. Which of the following is the feasible region (set) of the system of inequalities? +5 5 2 6 0 (a) Y 2 1-6 -5-4 -1 0-1 1 2 4 5 6 X
(b) Y 2-6 -5-4 1-1 0-1 1 2 4 5 6 X (c) Y 2-6 -5-4 1-1 0-1 1 2 4 5 6 X (d) Y 2 1-6 -5-4 -1 0-1 1 2 4 5 6 X 4
10. The corner points of the feasible region (set) of a linear programming problem are given in the following figure. What is the maximum value of the objective function +2 inthefeasibleregion? Y A(,1) 2 1 B(2, ) -4-1 0 1 2 4-1 C(2,) X (a) 8 (b) 10 (c) 0 (d) 10 11. A feasible set is described by the following inequalities: 0 0 2 +2 4 Which of the following is a corner point of the feasible region? (a) (1 1) (b) (0 2) (c) (2 1) (d) 8 2 12. Suppose that the constraints of a linear programming problem include the inequalities 0 and 0 The feasible (solution) set of the linear programming problem is restricted to which of the following quadrants. (a) I quadrant (b) II quadrant (c) III quadrant (d) IV quadrant 1. A company manufactures two ballpoint pens, silver and gold. The silver requires 1 min in a grinder and 7 mininabonder. Thegoldrequires2 min in a grinder and 10 min in a bonder. The grinder can be run no more than 1000 minutes per week and the bonder no more than 5600 minutes per week. The company makes a $4 profit oneachsilverpensoldand$7 on each gold. How many of each type should be made each week to maximize profits? (a) 0 silver and 500 gold (b) 50 silver and 00 gold (c) 00 silver and 50 gold (d) 560 silver and 0 gold 5
14. Suppose that $4 414 are invested at the simple interest rate of 10% What is the interest for 5 months? (Round your answer to two decimal places.) (a) $18 92 (b) $220 70 (c) $147 1 (d) $185 46 15. Find the amount of the monthly payment necessary to amortize the following: Loan amount $5 000; interest 6% per year compounded monthly; for 8 years. (Round your answer to two decimal places.) (a) $4 474 01 (b) $465 65 (c) $459 95 (d) $2 107 84 16. Jason took out a loan, to buy a new plasma TV, for years at the interest rate of 9 6% per year compounded monthly. His monthly payment is $57 74 After making 25 payments, he decided to pay off the remaining loan. What is the amount of his unpaid balance? (Round your answer to two decimal places.) (a) $1 799 89 (b) $605 68 (c) $1 0 61 (d) $82 0 17. Suppose that $6 000 are invested at 6 4% interest per year compounded quarterly for 5 years? How much interest is earned at the end of 5 years? (Round your answer to two decimal places.) (a) $8 41 86 (b) $2 41 86 (c) $2 241 86 (d) $495 60 18. Justin deposits $50 at the end of every week in an account that pays 7 8% interest per year compounded weekly. How much money is in the account at the end of 5 years? (Round your answer to two decimal places.) (a) $2 701 98 (b) $156 640 78 (c) $15 908 80 (d) $15 884 98 6
19. An account pays 9% interest per year compounded semiannually. What is the effective rate? (Round your answer to two decimal places.) (a) 9 8% (b) 9 00% (c) 9 20% (d) 9 1% 20. Let represents a false statement, represents a true statement, and represents a false statement. What is the truth value of the statement: ( ) ( ( )) (a) True (b) False (c) Not enough information to evaluate the statement. (d) None of these 21. Use DeMorgan s law to write the negation of the following statement. I did not pay my rent and I went to the football game. (a) I paid my rent and I did not go to the football game. (b) I did not pay my rent and I did not go to the football game. (c) I paid my rent or I did not go to the football game. (d) IdidnotpaymyrentorIdidnotgotothefootballgame. 22. What is the contrapositive of the statement: Orange juice contains vitamin C. (a) If it contains vitamin C, then it is not orange juice (b) If it does not contain vitamin C, then it is not orange juice. (c) If it contains vitamin C, then it is orange juice. (d) If it does not contain vitamin C, then it is orange juice 2. Which of the following is the truth table of the statement ( ) (a) (b) (c) ( ) ( ) ( ) 7
( ) (d) (e) None of the above 24. Determine whether the following statements are equivalent? ( ) and (a) Equivalent (b) Not equivalent (c) Not enough information to decide (d) None of the above 25. Determine if the following argument is valid. If Bill is a gambler, then he lives in Las Vegas. If Bill lives in Las Vegas, then Bill has a dog. Bill does not have a dog. Therefore, Bill is not a gambler. (a) The argument is valid. (b) The argument is not valid. (c) Not enough information to decide. (d) None of these 26. Let = { } Find the number of subsets of (a) 8 (b) 4 (c) 15 (d) 16 27. Let be sets. Which of the following is the Venn diagram of the set ( ) 0 A B A B C C (a) (b) A B A B C C (c) (d) 8
28. Let and be subsets of a universal set such that ( ) =50 ( ) =9 and ( ) =80 What is ( )? (a) 9 (b) 7 (c) 41 (d) 0 29. A survey of 150 students was done to find out the language classes they were taking. Let be the set of students taking Spanish, be the set of students taking French, and be the set of students taking Latin. The survey revealed the following information: ( ) = 45; ( ) = 55; ( ) = 40; ( ) = 12; ( ) = 15; ( ) = 2; ( ) =2 How many students were not taking any of these languages? (a) 60 (b) 58 (c) 68 (d) 19 0. A die is rolled twice. Find the event that the sum of the numbers rolled is either 4 or 5 (a) {(2 2) (2 )} (b) {(1 ) ( 1) (2 2) (1 4) (4 1) (2 ) ( 2)} (c) {(1 ) (1 4) (2 2) (2 )} (d) {(2 2) ( 1) ( 2) (4 1)} 1. Each letter of the word is written on a different card and the cards are placed in a bag. One card is drawn at random. What is the probability that the drawn card shows an (a) 1 4 (b) 11 1 (c) 11 (d) 11 2. A bag contains 5 red, 2 yellow, and 4 blue marbles. One marble is drawn at random. Find the odds in favor of drawing a blue marble. (a) 1 to 4 (b) 4 to 11 (c) 7 to 11 (d) 4 to 7 9
. Let and be events in a sample space such that Pr[ ] =0 40 Pr[ ]=0 6 and Pr[ ]=0 24 Find Pr[ ] (a) 1 00 (b) 0 76 (c) 0 52 (d) 0 9 (e) None of the above. 4. Find the number of ways to select five cards from a deck of 52 cards such that 2 are aces and are face cards. (Only picture cards are face cards.) (a) (4 2) (12 ) (b) (4 2) + (12 ) (c) (16 5) (d) (16 ) 5. Find the number of distinct permutations of the letters of the word (a) 9 916 800 (b) 277 200 (c) 18 600 (d) 69 00 6. A debate club contains 6 students from Arts and Sciences, students from Business Administration, and 2 from the Law school. To participate in the next debate competition, students are selected at random. Find the probability that 1 student is from Arts and Sciences and 2 arefromthelawschool. (Round your answer to four decimal places.) (a) 0 004 (b) 0 064 (c) 0 0424 (d) 0 66 10
7. At a Humane Society, 0% of the dogs are considered large, 45% are medium, and 25% are small. Some of the dogs have been raised as outside dogs, those that are not allowed inside the owner s house. The rest are allowed to come inside. The percents of animals in each category are summarized on the following tree. Large 0.50 outside 0.0 0.50 in or out 0.45 Medium 0.40 0.60 outside in or out 0.25 0.20 outside Small 0.80 in or out A dog is selected at random. What is the probability that the dog is a small outside dog? (a) 0 25 (b) 0 20 (c) 0 05 (d) 0 25 8. Refer to the tree diagram of Exercise 7. A dog is selected at random. If the dog selected is medium sized, what is the probability that it is an in or out dog? (a) 0 45 (b) 0 60 (c) 0 27 (d) 0 75 9. Of groundhogs living in a Midwestern state, 70% hibernate all winter. Forty percent are afraid of their own shadows, and 28% both hibernate and fear their own shadows. A groundhog is chosen at random. If the groundhog fears its shadow, what is the probability that it hibernates? (a) 0 40 (b) 0 70 (c) 0 28 (d) 0 42 11
40. Of groundhogs living in a Midwestern state, 70% hibernate all winter. Forty percent are afraid of their own shadows, and 28% both hibernate and fear their own shadows. Which of the following statements is true regarding the events the groundhog hibernates all winter and the groundhog does not fear its shadow.? (a) The events are mutually exclusive. (b) The events are independent. (c) The events are not independent. (d) The events are pairwise discreet. statements is true. 41. Of the outstanding bills at a local dentist s office, only 22% of those over 90 days overdue are expected to be paid. Six bills are chosen at random. What is the probability that 2 of these bills will be paid? (a) 0 021 (b) 0 5 (c) 0 048 (d) 0 7 42. Find the mean. Round your answer to the nearest tenth. Value Frequency 10 2 15 1 18 2 25 4 0 1 (a) 20 1 (b) 40 2 (c) 9 8 (d) 10 8 4. Find the median of the following data: 5 27 1 50 47 5 (a) 5 (b) 1 (c) 25 (d) 27 44. Find the total area under the standard normal curve between the -scores =0 60 and =1 98 (a) 2 58 (b) 1 8 (c) 0 2504 (d) 0 2504 12
45. Find the -score satisfying the following condition: 25 78% area is to the right of (a) 0 65 (b) 0 65 (c) 1 65 (d) 1 65 46. Suppose that a fair coin is tossed 500 times. Use a normal approximation to a binomial distribution to find the following probability: Less than or equal to 20 tosses are heads. (Round your answer to three decimal places.) (a) 0 07 (b) 0 050 (c) 0 041 (d) 0 042 47. Suppose that two percents of the MP players manufactured are defective. Find the probability that in a shipment of 10 000 MP players exactly 225 are defective. (a) 0 0057 (b) 0 004 (c) 0 0055 (d) 0 0060 1
Answers: FormA 1. B 2. A. B 4. C 5. A 6. B 7. B 8. D 9. D 10. A 11. D 12. A 1. C 14. A 15. C 16. B 17. C 18. D 19. C 20. A 21. C 22. B 2. C 24. B 25. A 26. D 27. A 28. A 29. B 0. B 1. B 2. D. C 14
4. A 5. B 6. B 7. C 8. B 9. B 40. B 41. E 42. A 4. D 44. C 45. A 46. C 47. A 15