A man has ​$380000 invested in three rental properties.One property earns

7.5​% per year on the​ investment, the second earns 8​% and the third earns 9​%.

The total annual earnings from the three properties is $ 32,000 and the amount invested at 9​%

equals the sum of the first two investments. Let x equal the investment at

7.5​%, y equal the investment at 8​%,and z represent the investment at 9​%.

A.Solve the system of equations to find how much is invested in each property.

Explain how you can solve the following problems using the QR factorization.

(a) Find the vector x that minimizes ||Ax − b1||^2 + ||Ax − b2||^2 . The problem data are the m × n matrix A and two m-vectors b1 and b2. The matrix A has linearly independent columns. If you know several methods, give the most efficient one.

(b) Find x1 and x2 that minimize ||Ax1 − b1||^2 + ||Ax2 − b2||^2 . The problem data are the m × n matrix A, and the m-vectors b1 and b2. The matrix A has linearly independent columns.

Consider the following second-order ODE,

y"+1/4y=0

with y(0) = 1 and y'(0)=0.

Transform this unique equation into a system of two 1st-order ODEs.

Solve the obtained system for t in [0,0.6] with h = 0.2 by MATLAB using Euler Method or Improved Euler Method.

Use Laplace transforms to solve the initial value problem

x′′+4x′+3x=t,x(0)=0,x′(0)=3,

where primes indicate derivatives with respect to t.

Solve y''-y'-2y=e^t using variation of parameters.

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