2. (a) Show that 341 is a composite number.
(b) Show that 341 is a pseudoprime to base 15.
(c) Show that 341 is not a strong pseudoprime to base 15.

Compute the reduced homology groups of the Mobius strip. (Hint: use homotopy invariance)

Recall that a sequence an is Cauchy if, given ε > 0, there is an N such that whenever m, n > N, |am − an| < ε.

Prove that every Cauchy sequence of real numbers converges.

Find solutions to the following difference equations:

• xn+2 − 4xn = 27n 2 , x0 = 1, x1 = 3

• xn+1 − 4xn + 3xn−1 = 36n 2 , x0 = 12, x1 = 0

• xn+1 − 4xn + 3xn−1 = 3n , x0 = 2, x1 = 13/2

• xn+2 − 2xn+1 + xn = 1, x0 = 3, x1 = 6

Find two integer pairs of the form (x,y) with |x|<1000 such that 22x+28y=gcd(22,28)

(x1,y1)=(

(x2,y2)=(

Orthogonally diagonalize the matrix by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) A = 5 0 0 0 1 3 0 3 1 (D, Q) = $$ Incorrect: Your answer is incorrect. Submission 2(0/1 points)Monday, November 25, 2019 10:01 PM CST Orthogonally diagonalize the matrix by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) A = 5 0 0 0 1 3 0 3 1 (D, Q) = $$ Incorrect: Your answer is incorrect. Submission 3(0/1 points)Monday, November 25, 2019 10:08 PM CST Orthogonally diagonalize the matrix by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) A = 5 0 0 0 1 3 0 3 1 (D, Q) = $$ Incorrect: Your answer is incorrect.

Find solutions to the following ODEs:

• y¨ − y˙ − 2y = t, y(0) = 0, y˙(0) = 1

• y¨ − 2 ˙y + y = 4 sin(t), y(0) = 1, y˙(0) = 0

• y¨ = t 2 + t + 1 (find general solution only)

• y¨ + 4y = t − 2 sin(2t), y(π) = 0, y˙(π) = 1

Consider the linear system of equations below
3x1 − x2 + x3 = 1
3x1 + 6x2 + 2x3 = 0
3x1 + 3x2 + 7x3 = 4
i. Use the Gauss-Jacobi iterative technique with x
(0) = 0 to find
approximate solution to the system above up to the third step
ii. Use the Gauss-Seidel iterative technique with x
(0) = 0 to find
approximate solution to the third step

Find the first two iterations of the SOR method with ω = 1.1 for the
linear system below using x
(0) = 0.
3x1 − x2 + x3 = 1
3x1 + 6x2 + 2x3 = 0
3x1 + 3x2 + 7x3 = 4

Find the general solution and solve the IVP

y''+y'-y=0, y(0)=2, y'(0)=0

Calculate the transform directly, by integration

a) L(5e^3t + 6cos4t + 3t^4)

b) L(5e^t sin4t)

c) L(5te^3t)

20. Give the values of x for which x > 500 lg x.

Note: lg is log base 2 (not ln).

Write a Fourier series equations solved example.

add bibliography reference.

Properties and characteristic of the Fourier series equations.

determine the range and domain of the following: the function k defined by k(x)= square root x-3