x2uxx --2xyuxy -- 3y2uyy = 0 u(x,1) = x, uy(x,1) = 1

Please show work

1) What combination of X and Y will yield the optimum for this problem?

Maximize Z = 10x+30y subject to (1) 4x + 6y < 12 and (2) 8x + 4y < 16

A) X=2.0, Y=0.0 B) X=1.5, Y=1.0, C) X=0.0, Y=2.0, D) X=3.0, Y=2.0 E) None

2) What combination of X and Y will provide a minimum for this problem?

Minimize Z = X + 3Y subject to: (1) 2X + 4Y > 12 and (2) 5X + 2Y > 10

A) X=6.0, Y=0.0 B) X=5.0, Y=0.0, C) X=1.0, Y=2.5, D) X=2.5, Y=1.0 E) None

Please show work

1. What is the subdivision and beat pattern of the following meter?

2. 6

3. 4

4. Simple quadruple

5. Compound Triple

6. Simple Triple

7. Compound Duple

8. What is the beat unit of the previous meter?

9. Quarter Note

10. Half Note

    Dotted Quarter Note

    Dotted Half Note

1. Simplify each of the following sets as much as possible.

a) ((? ∪ ?^?)^? ∩ (? ∩ ∅^?))^?

b) (ℤ ∩ ℚ⁺)^? ∩ ℤ_?

_{2.  Determine the cardinality of each set:}
a) ((ℂ − ℝ) ∩ ℕ) ∪ (ℤ⁺ − ℕ)
b) ?(ℤ_?^? ∩ ℤ_??)

1. (3 pts) Solve the initial value problem
   25y′′−20y′+4y=0, y(5)=0, y′(5)=−e2.

2. (3 pts) Solve the initial value problem
   y′′ − 2√2y′ + 2y = 0, y(√2) = e2, y′(√2) = 2√2e2.

3. Consider the second order linear equation t2y′′+2ty′−2y=0, t>0.

   1. (a) (1 pt) Show that y1(t) = t−2 is a solution.

   2. (b) (3 pt) Use the variation of parameters method to obtain a second solution and a general solution.

solve the wave equation utt=4uxx for a string of length 3 with both ends kept free for all time (zero Neumann boundary conditions) if initial position of the string is given as x(3-x), and initial velocity is zero.

The equation 2x₁-3x₂=5 defines a line in R2.

a. Find the distance from the point w=(3,1) to the line by using projection.
b. Find the point on the line closest to w by using the parametric equation of the line through w with vector a.

consider “COLLEGEOFENGINEERING”

a) How many permutations are there total?

b) How many permutations start and end with vowels?

c) How many permutations do NOT have consecutive vowels in them?

d) How many permutations contain the vowels in order (all Es before All Is before all Os)?

e) How many permutations contain the substring "GINGER"?

1. Which of the following maps define an isomorphism of binary structures? Explain. (a) The function (R, +) → (R, +) given by x → x 2 (b) The function (R>0, ·) → (R>0, ·) given by x → x 2 (c) The function (R>0, +) → (R>0, +) given by x → 2x 2. For each of the following, prove or disprove that it is a group. If it is a group, prove or disprove that it is abelian. (a) (Q +, ·) (b) (R \ {0}, ∗), where a ∗ b = ab 2 3. Assume that φ : (S, ∗) → (S 0 , ∗ 0 ) is an isomorphism of binary structures, and that ∗ is associative. Prove that ∗ 0 is associative. 4. Consider the set H = a −b b a ∈ M2(R) : a 6= 0 or b 6= 0 , with matrix multiplication as the operation. (a) Show that the operation is closed. (b) Define the function φ : H → C ∗ (where C ∗ is the group C \ {0} with multiplication as the operation) by φ : a −b b a 7→ a + bi. Show that φ is an isomorphism of binary structures. (c) Explain briefly why we can now conclude that H is a group. 5. Consider the set V = a 0 0 b ∈ M2(R) : a, b ∈ {1, −1} , with the operation · (matrix multiplication). Note that (V, ·) is a group (you do not need to prove this). Prove that (V, ·) is not isomorphic to U4.

3. Suppose that a divide and conquer algorithm for multiplication of n x n matrices is found such that it requires 6 multiplications and 31 additions of n/2 x n/2 submatrices. Write the recurrence for the running time T(n) of this algorithm and find the order of T(n).

(1) Recall on February 6 in class we discussed e 0 + e 2πi/n + e 4πi/n + · · · + e 2(n−1)πi/n = 0 and in order to explain why it was true we needed to show that the sum of the real parts equals 0 and the sum of the imaginary parts is equal to 0.

(a) In class I showed the following identity for n even using the fact that sin(2π − x) = − sin(x): sin(0) + sin(2π/n) + sin(4π/n) + · · · + sin(2(n − 1)π/n) = 0 Do the same thing for n odd (make sure it is clear, at least to yourself, why the argument is slightly different for n even and n odd).

(b) Using the identity cos(x) = − cos(x + π), show that cos(0) + cos(2π/n) + cos(4π/n) + · · · + cos(2(n − 1)π/n) = 0 for n even.

(c) Why does the same proof not work for n odd ? Show and explain what goes wrong for the example of n = 3.

Explain how the rank of a matrix and existence and uniqueness of solutions of “Systems of Linear Equations” are related

.

Explain how Eigenfunctions”, “Eigenvalues” and “Orthogonality” terms and concepts are defined in Matrix algebra.

Parts A-D

A)On average, a student takes 130 words/minute midway through an advanced court reporting course at the American Institute of Court Reporting. Assuming that the dictation speeds of the students are normally distributed and that the standard deviation is 10 words/minute, find the probability that a student randomly selected from the course can take dictation at the following speeds. (Round your answers to four decimal places.)

(1) more than 150 words/minute______


(2) between 100 and 150 words/minute________


(3) less than 100 words/minute_________

B)The tread lives of the Super Titan radial tires under normal driving conditions are normally distributed with a mean of 30,000 mi and a standard deviation of 5000 mi. (Round your answers to four decimal places.)

(1)What is the probability that a tire selected at random will have a tread life of more than 24,500 mi?_________


(2)Determine the probability that four tires selected at random still have useful tread lives after 24,500 mi of driving. (Assume that the tread lives of the tires are independent of each other.)_________

C)Let Z be the standard normal variable. Find z if z satisfies the given value. (Round your answer to two decimal places.)

P(Z < z) = 0.1210

(1)z=_________

D)Let Z be the standard normal variable. Find z if z satisfies the given value. (Round your answer to two decimal places.)

P(Z > z) = 0.9878

(1)z=_______

Include an example of an appropriate matrix as you justify your responses to the following questions.

1. Suppose a linear system having six equations and three unknowns is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely solutions?

2. Suppose that a linear system having three equations and six unknowns is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely solutions?

3. Suppose that a linear system is consistent and has a unique solution. What can you guarantee about the pivot positions in the augmented matrix?