Math 1324 – Final Exam Review Page 1

Math 1324 Final Exam Review

Test instructions

Date and Time: May 10th, 3:10 PM – 5:10 PM

What is covered? Everything we’ve learned

Number of questions: 30 Multiple-Choice Questions

You can use a graphing calculator on the final exam.

Remember the make-up policy: NO MAKE UPS!

1. Which of the following matrices are in row-reduced echelon form?

I. [1 0 000

10

01

|−2−63

]

II. [1 0 −200

00

10

|−630

]

III. [1 10 0

|−20

]

IV. [1 1 00 2 0

|−64

]

2. Given that the augmented matrix in row-reduced echelon form below is equivalent to the augmented

matrix of a system of linear equations. Determine whether the system has a solution and find the

solution(s) to the system, if they exist.

[1 0 00 1 10 0 0

001

|8

−3−1

]

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3. Use the Gauss-Jordan elimination process on the following system of linear equations to find the

value of z.

−𝑥 + 3𝑦 + 𝑧 = 16𝑥 − 5𝑦 + 3𝑧 = 2

−2𝑥 + 5𝑦 − 3𝑧 = 6

4. Use the Gauss-Jordan elimination process to solve the following system of linear equations.

6𝑥 + 2𝑦 − 12𝑧 = −23𝑥 + 𝑦 − 6𝑧 = −2

Math 1324 – Final Exam Review

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5. Calculate the following if possible

−5 [−8 3 5−3 3 5

] + 9 [5 −8 6

−2 7 4]

6. Solve for z.

(𝑥 −58 −26

−63 𝑧 −1810 8 25

) + 2 (−9 4 39 10 4

−9 𝑦 10) = −5 (

2 10 49 8 2𝑤 −4 −9

)

7. Let A = [4 −10

−6 −2−9 8

] and B = [−4 45 −2

]. Compute the product AB, if possible.

Math 1324 – Final Exam Review

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8. A certain academic department at a local university will conduct a research project. The department

will need to hire graduate research assistants and professional researchers. Each graduate research

assistant will need to work 26 hours per week on fieldwork and 14 hours per week at the university's

research center. Each professional researcher will need to work 15 hours per week on fieldwork and 25

hours per week at the university's research center. The minimum number of hours needed per week for

fieldwork is 154 and the minimum number of hours needed per week at the research center is 140.

Each research assistant will be paid $218 per week and each professional researcher will be paid $499

per week. Let x denote the number of graduate research assistants hired and let y denote the number of

professional researcher hired. The department wants to minimize cost. Set up the Linear Programming

Problem for this situation.

9. Solve the following linear programming problem:

Maximize 𝑃 = 8𝑥 + 4𝑦 subject to {

4𝑥 + 𝑦 ≤ 42𝑥 + 5𝑦 ≤ 10

𝑥 ≥ 0𝑦 ≥ 0

Math 1324 – Final Exam Review

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10. Convert the inequalities in the following linear programming problem into equations by adding

slack variables.

Maximize 𝑧 = 3𝑥1 + 2𝑥2 + 𝑥3

Subject to: 2𝑥1 + 𝑥2 + 𝑥3 ≤ 150

2𝑥1 + 2𝑥2 + 8𝑥3 ≤ 200

2𝑥1 + 3𝑥2 + 𝑥3 ≤ 320

With 𝑥1 ≥ 0, 𝑥2 ≥ 0, 𝑥3 ≥ 0.

11. Find the solutions that can be read from the simplex tableau given below.

12. Find the basic variables from the simplex tableau given below.

Math 1324 – Final Exam Review

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13. Find the dual Problem.

Minimize: 𝑤 = ____________________

Subject to: ________________________

________________________

________________________

with 𝑦1 ≥ 0, 𝑦2 ≥ 0, 𝑦3 ≥ 0

14. Congratulations! You were the 10th caller on the KMTH morning show and you just won $6,000.

After you calm down, you decide to put the money into a bank account so that you will have even more

money for a trip to Europe. The bank tells you that they will pay 4% per year compounded monthly.

How much money will you have for your trip in 8 years.

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15. Jennifer bought a brand-new treadmill on a credit plan at BeFit Exercise Equipment. BeFit will

charge her 18% per year compounded monthly. If her monthly payments will be $150 for 2 years, how

much was the original price of the treadmill?

16. You borrowed $11,000 from your bank to build a small cabin on your property. The bank will

charge 10% per year compounded quarterly. You decide to pay off this loan in 3 years by making

quarterly payments. How much are your quarterly payments?

17. Esther pays $32 per month for 5 years for a car. She made a down payment of $4,500. If the loan

costs 7.1% per year compounded monthly, what was the cash price of the car?

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18. Given the following Venn diagram, find 𝑛(A ∩ B′).

19. Of the cars sold during the month of July, 95 had air conditioning, 102 had automatic transmission,

and 73 had power steering. 7 cars had all three of these extras. 25 cars had none of these extras. 21 cars

had only air conditioning, 55 cars had only automatic transmissions, and 35 cars had only power

steering. 9 cars had both automatic transmission and power steering. How many cars had exactly two

of the given options?

Math 1324 – Final Exam Review

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20. How many different license plates can be made if each license plate is to consist of 3 letters,

followed by 2 digits, and no letters or digits may repeat?

21. After a bride has interviewed 7 DJs to play at her wedding, her fiancé tells her she needs to narrow

it down to 2 DJs. In how many ways can she rank the 2 DJs?

22. In how many ways can a Math team of 9 students be chosen from a Math Club which consists of 19

seniors and 11 juniors if the team must consist of 5 seniors and 4 juniors?

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23. A house organization needs to make up a 6 member fund-raising committee. The organization has

10 accounting majors and 7 finance majors. In how many ways can the fund-raising committee be

formed if at most 1 accounting major is on the committee?

24. Given P(A) = 0.37, P(B) = 0.42 and P(A ∩ B′) = 0.17. Find P(A ∩ B)′.

25. A classroom of children has 16 boys and 20 girls in which five students are chosen to do

presentations. What is the probability that at least four boys are chosen?

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26. 7 black balls and 12 white balls are placed in an urn. Two balls are then drawn in succession. What is the

probability that the second ball drawn is a white ball given the first ball is white? The second ball is drawn

without replacing the first ball.

27. Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards. What is

the probability that the first card drawn is an ace and the second card drawn is a king?

28. Find the expected value of the random variable X having the following probability

distribution.

x -4 -3 1 5 6

P(X=x) 1/9 4/9 1/18 2/9 1/6

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30. Consider the following binomial experiment. The probability that a green jelly bean is chosen at random

from a large package of jelly beans is 1/8. Sally chooses 10 jelly beans, what is the probability that at most 2

will be green jelly beans?

31. Consider the following binomial experiment. A newspaper publisher claims that 64% of the people

in a certain community read his newspaper. Doubting the assertion, a competitor randomly surveys 292

people in the community. Find the standard deviation of this experiment.

Math 1324 – Final Exam Review