Solve each system. Express the solution set using vectors. Identify a particular solution and the solution set of the homogeneous system.

(a) 3x + 6y = 18

x + 2y = 6

(b) x + y = 1

x − y = −1

(c) x₁ + x₃= 4

x₁ − x₂ + 2x₃ = 5

4x₁ − x₂ + 5x₃ = 17

(d) 2a + b − c = 2

2a + c = 3

a − b = 0

(e) x + 2y − z = 3

2x + y + w = 4

x − y + z + w = 1

(f) x + z + w = 4

2x + y − w = 2

3x + y + z = 7

Reduce the following argument into an argument form, and then perform the truth table test for validity. Create and label the entire truth table.
If your program contains a syntax error or a semantic error, then your program will fail to compile.
Your program contains a syntax error and a semantic error.
Therefore, your program will not fail to compile.

The Tiny Toy Company makes three types of new toys: the tiny tank, the tiny truck, and the tiny turtle. Plastic used in one unit of each is 1.5, 2.0 and 1.0 pounds, respectively. Rubber for one unit of each toy is 0.5, 0.5, and 1.0 pounds, respectively. Also, each tank uses 0.3 pounds of metal and the truck uses 0.6 pounds of metal during production. The average weekly availability for plastic is 16,000 pounds, 9,000 pounds of metal, and 5,000 pounds of rubber. It takes two hours of labor to make one tank, two hours for one truck, and one hour for a turtle. The company allows no more than 40 hours a week for production (priority #1). Finally, the cost of manufacturing one tank is $7, 1 truck is $5 and 1 turtle is $4; a target budget of $164,000 is initially used as a guideline for the company to follow.

a) Minimize over-utilization of the weekly available supply of materials used in making the toys and place twice as much emphasis on the plastic (priority #2)

b) Minimize the under and over-utilization of the budget. Maximize available labor hour usage (priority #3).

Formulate the above decision problem as a single linear goal program. Clearly identify your achievement vector (i.e., hierarchy of priority levels for the goals). Do not solve.

Find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra.

y⁽⁴⁾ + 2y'' + y = 11 cos(x) − 12x sin(x)

Use the method of variation of parameters to find the general solution of the differential equation

y''+6y'+5y = 7e^(2x)

describe group homomorphisms from Q8 into Z8.

1. A 60 kg individual needs to work at a MET level of 8 to reach her target heart rate. If the treadmill speed is set at 3.5 mph, what grade is needed? If she exercises for 20 min. at this intensity, how many kcal will she expend?

5A

Research and share two or more real life applications of radial, rotational, or gradient fields.

5B

Explain the concept of divergence and provide an example showing how to calculate the divergence of the vector field chosen in your example.

1. Write notes on Multicollinearity
2. 1. define what the issue is
   2. explain whether, or under what circumstances, it causes a problem for inference in OLS regression
3. 3. explain how it can be detected (if it can)
   4. state what action, if any, you should take to deal with the issue

1. Write notes on omitted variable bias.  
   
2. 1. define what the issue is
   2. explain whether, or under what circumstances, it causes a problem for inference in OLS regression
3. 3. explain how it can be detected (if it can)   
   4. state what action, if any, you should take to deal with the issue.

PLEASE EXPLAIN WHY YOU CHOOSE EACH ANSWER

1). How many distinct ways can a President, Vice President, Secretary and Treasurer be selected from a group of 10 people if no one can hold more than on position?

A). P(10,4)

B). 10 choose 4

C). 10^4

D). 4^10

E). 13 choose 10

F). None of these

2). How many shortest lattice paths are there from (0,0) to (10,4)

A). P(10,4)

B). 10 choose 4

C). 10^4

D). 4^10

E). 13 choose 10

F). None of these

3). At a movie theater with 4 different movies, how many ways can 10 people select a show? They do not have to all go to the same shoe but several poeple can go to the same show.

A). P(10,4)

B). 10 choose 4

C). 10^4

D). 4^10

E). 13 choose 10

F). None of these

The tarot minor arcana deck has 56 cards in four suits (swords, batons, coins, and cups), each having 10 numbered cards and 4 court cards (king, queen, knight, and page). A “spread” is a draw of 9 cards. What is the probability of getting a spread of all court cards?

Creating a Hypothetical Budget – Portfolio Activity

You are paid $11.75/hr. You work 40 hr/wk. Your deductions are FICA (7.65%), Federal tax withholding (10.75%), and state tax withholding (7.5%)

1. Assuming you budget a month as 4 weeks, how much is your total realized income, fixed expenses, and discretionary expenses.
2. If you are able to work 20 hours of overtime the next month and are paid 1.5 times your regular rate, how does this change your budget for that month?
3. How much can you put towards savings each month if you eliminated your discretionary expenses?

Be sure to include in your response:

• A detailed budget showing income and expenses for the two different months.
• Answers to the original questions.
• An explanation for how you calculated the first 40 hours of income and then the additional 20 hours of overtime.

Month 1

Month 2

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1. The value of a European put option must satisfy the following restriction:

??0 ≥????−???? −??0

where ??₀ is the current put price, ??₀ is the current price of the underlying stock, ?? is the exercise price, ?? > 0 is the annualised continuously compounded risk-free rate, and ?? is the time till expiration. Prove by contradiction that the above arbitrage restriction must hold, i.e. show that if the condition does not hold, there is an arbitrage opportunity