The Janie Gioffre Drapery Company makes three types of draperies at two different locations. At location I, it can make 10 pairs of deluxe drapes, 20 pairs of better drapes, and 13 pairs of standard drapes per day. At location II, it can make 20 pairs of deluxe, 50 pairs of better, and 6 pairs of standard per day. The company has orders for 3000 pairs of deluxe drapes, 6300 pairs of better drapes, and 1800 pairs of standard drapes. If the daily costs are $450 per day at location I and $600 per day at location II, how many days should Janie schedule at each location to fill the orders at minimum cost?

location I days ?

location II days?

Find the minimum cost. $

Show that the integral ( over a volume) of the curl of the vector A is equal to the integral over a closed surface (containing the volume) of A x da

Let x ∈ Rⁿ be any nonzero vector. Let W ⊂ R^nxn
consist of all matrices A such that Ax = 0. Show that W is a subspace and find its dimension.

Check that the following differential equation is not exact. Find an integrating factor that makes it exact and solve it.

ydx + (3 + 3x-y) dy = 0

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Solve the initial value problem:

(cos xsen x-xy ^ 2) dx + (1-x ^ 2) ydy = 0 if y (0) = 2

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Solve the initial value problem:

y ^ (2) cosx dx + (4 + 5ysenx) dy = 0; y (π / 2) = 1

Find the particular integral of the following differential equations.(Explain each step clearly)

(a) d²y/dx² + y = (x + 1) sin x. show that the answer is yp(x) = − 1/8 [ (2x² + 4x − 1) cos x − (2x + 2) sin x ]

(Hint:In this case, we substitute sin αx or cos αx with e^iαx then use the shift operator. In the case of sin αx we extract the imaginary part.)

How many permutations of the letters ABCDEFGHIJKLM do not contain the strings “BAD” or “DIG” or “CLAM” consecutively? (Hint: Inclusion-exclusion and subtraction).

**I KNOW THE ANSWER IS NOT 13! - 11! - 11! - 10! ** - please do not give that as one.

2. (a.) Consider [3]₁₂ in Z₁₂. Is [3]₁₂ a zero divisor of Z₁₂ and does [3] have a multiplicative inverse in Z₁₂? Justify your answer.

(b.) Prove that [3]₁₀ has a multiplicative inverse in Z₁₀ and determine the order of [3] in Z^x₁₀. Justify your answer.

(c.) Prove that [6]₈ is a nilpotent element of Z₈. Justify your answer.

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.