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Arizona State University Math 119 Finite Mathematics Final Exam A

Dr. S. Takahashi

Your Predicted Score: _____________

Your Actual Score: _____________

Read the following before you take this test.

1) Maximum total points for this test is 150.

2) You have 110 minutes to complete. Your score is reduced by one point per each minute exceeded.

3) Show all of your work. No work [pic] No credit

4) No need to show work for true-or-false questions.

5) If the problem has the designation “No calculator”, follow that policy.

6) If you are row-reducing the matrix, make sure to indicate the operation applied for each step.

7) Make sure to define events wherever applicable.

8) For finance calculations, round your answers to the nearest cent.

9) You are not allowed to use calculators for those problems indicated as such.

10) Simplify solutions as far as possible. Do not leave solutions in symbolic forms, such as[pic].

11) Use only Casio CFX-9850GB, TI-81, 83, 86 or ordinary non-graphing calculators, if needed.

12) Use scratch paper provided if you need one. Attach to the test.

13) Make sure you indicate your solutions by circling or boxing your final answers.

14) Before you start, make sure you have all questions.

15) No rubber-necking.

Good luck.

Honor Statements: (Standard ASU math dept. version)

By signing below you confirm that you have neither given nor received any unauthorized assistance on this exam. This includes any use of a graphing calculator beyond those uses specifically authorized by the Mathematics Department and your instructor. Furthermore, you agree not to discuss this exam with anyone until the testing period is over. In addition, your calculator’s program memory and menus may be checked at any time and cleared by Mathematics Department instructor.

Signature Date

Show all work, except for True or False questions. Throughout [pic] probability of an event [pic].

1. True or False: (2 points each)

1. Every matrix has a unique row echelon matrix associated with it.

______________

2. Let A be an augmented matrix for the linear system. If number of nonzero

rows of reduced row echelon form of A is less than number of its column,

then the system must possess infinite solutions.

______________

3. In linear programming, minimizing an objective function [pic] in the standard

feasible region is equivalent to maximizing [pic]in the same feasible region.

______________

4. An equiprobable sample space is a sample space in which all events of the

sample space have equal probabilities.

______________

5. You can generate two mutually exclusive events that are also independent.

______________

カ) I, the examinee, have learned a lot in this class.

______________

2. Construct the augmented matrix for the following system and solve the system using: (pts. total)

1. Gaussian Elimination Method

2. Gauss-Jordan Method.

Note: For both parts, make sure that you indicate a row operation you employed for each step.

No Calculator.

[pic]

3. Consider two matrices [pic] and [pic] (points each)

1) Calculate [pic] if possible. If not, then provide an argument why not. No calculator.

2) Calculate[pic] if it exists. If it does not exist, state the reason(s) why not. No calculator.

4. Fill the blanks. (points each)

ア） Dimension or size of the matrix [pic] is ______________________.

イ） Matrix [pic] is said to be ________________ if [pic] exists.

ウ） Let [pic]be a set with [pic]. Then [pic] _________________ , where [pic]

is the power set of [pic].

エ） The effective annual yield for 8% compounded semiannually is ____________________.

オ） Suppose sets [pic]and [pic]are such that cardinalities of [pic], [pic] and [pic] are 11, 6 and 30 respectively. Then the cardinality of [pic] is ______________________.

5. In the laboratory, you are asked to mix three HCL solutions with distinct concentrations. Their concentrations are: 10%, 20% and 50%. Calculate the number of liters of each that you need to mix so that you will get 100 liters of 25% HCL solution. Make sure to give all possible solutions. (10 points)

(2.3 LN)

　

6. A Classic company produces two types of hand made baseball gloves. An Infielder Model requires 2 hours of cutting leather, 4 hours forming and 3 hours of stitching. An Outfielder Model requires 1 hours of cutting, 4 hours of forming and 3 hours of stitching. 30 hours are allocated for cutting, 50 hours for forming and 40 hours for stitching. An Infielder Model generates a profit of 150 dollars whereas an Outfielder Model generates a profit of 120 dollars. The Classic company would like to maximize its profit.

ア) Introduce two independent variables for this problem.

2. Introduce the objective function for this problem.

3. Write down all constraints for this problem.

4. Sketch a graph of the feasible region.

5. Calculate the maximum profit. Also calculate the number of gloves of each type the

company needs to manufacture in order to generate this maximum profit.

7. You are to solve the following problem using the Simplex Method. Answer the following questions.

(20 points total)

Maximize: [pic]

Subject to constraints: [pic], [pic], [pic]

1. Introduce slack variables and convert above into a system of three equations. Make sure to include inequalities that indicate all decision variables([pic]) and slack variables are non-negative.

2. Set up the initial simplex tableau for this problem. Identify the pivot for this tableau by circling it.

3. Apply pivoting to find the maximum value of [pic]. Also write down values of all basic and non-basic variables with which[pic] attains its maximum.

8. Suppose the price of the house you have purchased was $180,000 and you made a down payment of $50,000. You are able to amortize the balance at 7% for 30 years from the bank. (points total)

1) Calculate the monthly mortgage payment.

2) Calculate the total interest.

3) Calculate the equity of the house after 20 years.

9. Answer the following two questions.

a) Suppose you deposit $300 monthly for 4 years at an annual rate of 4 % compounded monthly. How much do you have in 4 years? (points)

b) Suppose you deposit $500 monthly for 5 years at an annual rate of 4%. You want to have $25,000 at the end. How many years do you need to invest in order to attain your goal? Round your answer to one-tenth of a year.

10. Suppose a class of Mat 119 consists of 35 % freshmen, 25 % sophomores, 20 % juniors and 20 % seniors.

Suppose also that 5 %, 30 %, 60 % and 70 % of students in each class passed the placement tests, respectively. Answer the following questions.

1. Calculate the total probability for the event that a student passed the placement test.

2. If a particular student passed the placement test, calculate the probability that the student is sophomore.

11. Police says that the odds are 5 to 2 that a tricycle stolen in Tempe will be retrieved. An actuary wants to determine the probability that exactly 7 out of 10 tricycle stolen in Tempe will be recovered for tricycle insurance policy. Calculate the probability. Assume that each thievery is independently performed by a distinct thief and thieves do not share any information that would alter the probability.

12. Suppose that the odds for getting a jackpot are 5 to 7. When you win, you keep the money you paid to play and you also get $20. How much are you willing to pay to play so that the game is fair?

13. Suppose the warehouse contains 5 malfunctioning computers and 11 functioning ones. Calculate the expected number of malfunctioning computers, if you are to bring 5 computers, one computer at a time, without bringing each back to the warehouse.

14. Mathematically determine whether the events [pic]and [pic] below are independent or not.

Your are tossing two fair dice. Events are:

[pic]: The first die shows a 5.

[pic]: The second die shows a 4.

[pic]: The sum of the 2 die is 7.

15. In world series, American league and National league teams play at most seven games to decide the winner. They cease to play once one of the teams wins four games. Calculate the number of possible sequence that world series can generate. For instance, a sequence ANNNAN means National league won the world series by winning the second, the third, the fourth and the sixth games.

15. You have learned both geometric approach and simplex method to solve linear programming problems. In what circumstance are you are bound to use simplex method instead of geometric approach? Provide the most obvious circumstance. (5 points)

Bonus Question (5 points):

Discuss the origin of the name “Simplex Method”, or why we would call the method “Simplex Method”. It would help to discuss what it means by an [pic] in an [pic]dimensional Euclidean space (i.e., [pic]), explaining why using the word “Simplex” is appropriate for the method.

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