Show that x^8 ≡ 2 ( mod 13) has no solutions.

1)
a.   Which of the following sets are the empty set?
i.   { x | x is a real number and x2 – 1 = 0 }
ii.   { x | x is a real number and x2 + 1 = 0 }
iii.   { x | x is a real number and x2 = -9 }
iv.   { x | x is a real number and x = 2x + 1 }

b.   Let A = {1, 2, 3, 4, 5}. Which of the following sets are equal to A?
i.   {4, 1, 2, 3, 5}
ii.   {2, 3, 4}
iii.   { x | x is an integer and x2 <= 25 }
iv.   { x | x is a positive rational number and x <= 5 }

Consider the initial value problem

mx′′+cx′+kx=F(t),   x(0)=0,   x′(0)=0
modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=80cos(8t) Newtons.

Solve the initial value problem.

x(t)=

Determine the long-term behavior of the system. Is limt→∞x(t)=0? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t.

For very large positive values of t, x(t)≈

A zip code is a string of five digits (0,1,2,3,4,5,6,7,8,9). How many zip codes are there subject to the following restrictions?
(1) Only even digits are allowed.
(2) The first digital is even and the last digit is odd
(3) Digits cannot be repeated
(4) 0 appears exactly four times
(5) 0 appears at at least once

Solve the following discrete mathematics problem above. Show all work/explanations

Write a formal proof to prove the following conjecture to be true or false.

If the statement is true, write a formal proof of it. If the statement is false, provide a counterexample and a slightly modified statement that is true and write a formal proof of your new statement.

Conjecture: Let w, x, y, and z be single-digit numbers. The 4-digit number wxyz* is divisible by 9 if and only if 9 divides the sum w + x + y + z.

* I don't mean the product of the of these numbers. I mean a four-digit number like 7,235 where w = 7, x = 2, y = 3, and z = 5 are the digits.

Solve the initial value problem below using the method of Laplace transforms.

y'' - 4y' + 8y = 5e^t

y(0) = 1

y'(0) = 3

Give a real-world example of the inclusion/exclusion principle that involves at least two finite sets. Specify values for three of the following four values, the size of the first set, the set of the second set, the size of the union and the size of the intersection. Apply the inclusion/exclusion principle to determine that value of the one value that you did not specify.

A least squares adjustment is computed twice on a data set. When the data is minimally constrained with 20 degrees of freedom, a reference variance of 0.69 is obtained. In the second run, the fully-constrained network also having 20 degrees of freedom has a reference variance of 1.89. The a priori estimate for the reference variance in both adjustments is 1;

a. Is the minimally-constrained adjustment reference variance statistically equal to 1 at a 0.05 level of significance?  

b. Is the fully-constrained adjustment reference variance statistically equal to 1 at a 0.05 level of significance?

c. Are the two variances statistically equal at a 0.05 level of significance?

d. Is there statistical reason to be concerned about the presence of errors in either the control or the observations?

Write a poem related to Calculus 2 content that students cover in that class!

Prove that isomorphism is an equivalent relation on the set of all groups.

A power shovel with a dipper of two cubic yard capacity has a standard operating cycle time of 80 seconds. The excavated material which has a swell factor of 05 will be disposed by a dump truck with an 8 cubic yard capacity at a dump site 6 miles away. The average speed of the dump truck is 30 mph and the dumping time is 40 seconds. Find the daily standard production rates of the power shovel and the dump truck if both are operated 8 hours per day. Determine also the number of trucks needed daily to dispose of the excavated material.

Formulate the definitions of linear dependence and independence of a set of k vector functions f₁(x), ... , f_k(x) continuous on an interval I.

Solve the following problem using the simplex method. If the problem is two dimensional, graph the feasible region, and outline the progress of the algorithm.

Max               Z = 5X₁ + 3X₂ + 2X₃

Subject to    4X₁ + 5X₂ + 2X_{3 +} X4≤ 20

                     3X₁ + 4X₂ - X_{3 +} X4≤ 30

                      X₁, X2, X_3, X₄ ≥ 0

Assuming we have a case of influenza. Suppose the total cost of providing viraflu is $100 and the total cost of providing supportive care is $10. Suppose further that viraflu will result in a 0.5 QALY per person treated and providing supportive care alone results in 0.1 QALY.

1. What is the incremental cost-effectiveness of providing viraflu to persons with influenza relative to providing supportive care alone?

2. Suppose that the total cost of vaccinating an individual is $150 and that vaccination results in the gain of 0.75 QALY per person. Calculate the average cost-effectiveness of influenza vaccination and the incremental cost-effectiveness of influenza vaccination relative to treatment?