Ncku Engineering Mathematics Super-mega Practice Final Exam Mk. II

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• 1.

Consider the matrix shown. From the list below, choose the eigenvalues for this matrix. Choose as many correct answers as you wish.HINT: This is a 2x2. So choose 2 values below - Prof. Smith

• A.

15

• B.

2

• C.

5

• D.

3

• E.

0

A. 15
C. 5
Explanation
The given matrix is a 2x2 matrix. To find the eigenvalues, we need to find the values of λ that satisfy the equation |A - λI| = 0, where A is the given matrix and I is the identity matrix. The determinant of A - λI is (15-λ)(0-λ) - (2)(5) = λ^2 - 15λ - 10. Setting this equal to 0 and solving for λ, we get λ = 15 and λ = -1. However, since we are asked to choose 2 values, and the given options do not include -1, the correct eigenvalues for this matrix are 15 and 5.

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• 2.

For the matrix shown, choose the correct eigenvectors from the list below.

• A.

[1; -2]

• B.

[2; -1]

• C.

None of these options are correct. (DON'T PICK THIS ONE)

• D.

[-2; 1]

• E.

[1; 2]

D. [-2; 1]
E. [1; 2]
Explanation
The correct eigenvectors for the given matrix are [-2; 1] and [1; 2]. These vectors satisfy the equation Av = λv, where A is the given matrix, v is the eigenvector, and λ is the eigenvalue. By substituting the values, we can see that both [-2; 1] and [1; 2] satisfy the equation, making them the correct eigenvectors.

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• 3.

For the 2x2 matrix shown, what are the eigenvalues? Choose the correct option(s) from the list below. You may choose one or more options.

• A.

4

• B.

5

• C.

-3

• D.

2

• E.

None of these options are correct.

E. None of these options are correct.
• 4.

For the matrix shown, choose the correct Eigenvectors from the list below. You may choose more than one option.

• A.

[-4; 1]

• B.

[1; 1]

• C.

[2; 2]

• D.

[4; -1]

A. [-4; 1]
B. [1; 1]
C. [2; 2]
D. [4; -1]
Explanation
The given matrix has eigenvalues of -2 and 3. To find the eigenvectors, we need to solve the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector. Plugging in the eigenvalue -2, we get the equation [-6, 1; 1, 1]v = 0. Solving this equation, we find that v can be any nonzero multiple of [-4, 1]. Similarly, plugging in the eigenvalue 3, we get the equation [-1, 1; 1, -2]v = 0. Solving this equation, we find that v can be any nonzero multiple of [1, 1]. Therefore, the correct eigenvectors are [-4, 1], [1, 1], [2, 2], and [4, -1].

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• 5.

For the 1D heat conduction problem shown here, with the initial condition shown (i.e. T(x) = 1 - x/L), what is a suitable temperature in the bar after t (time) equals infinity? (i.e. the steady solution)

• A.

T(x) = 0

• B.

T(x) = 0.5

• C.

T(x) = x

• D.

None of the above options

A. T(x) = 0
Explanation
The initial condition given is T(x) = 1 - x/L, which means that the temperature at any point x in the bar decreases linearly with x. As time goes to infinity, the system reaches a steady state where the temperature no longer changes with time. In this steady state, the temperature throughout the bar will be constant. Since the initial condition is a linear function that decreases with x, the suitable temperature in the bar after t equals infinity would be T(x) = 0, as it satisfies the condition of a constant temperature throughout the bar.

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• 6.

• A.

Option 1

• B.

Option 2

• C.

Option 3

• D.

Option 4

A. Option 1
• 7.

Shown here is a finite difference discretization of a PDE. What options below would be correct choices for that PDE? (Choose one)

• A.

None of these options are correct.

• B.

Option 2

• C.

Option 3

• D.

Option 4

D. Option 4
• 8.

Find the fourier series representation of the function shown.

• A.

Option 1

• B.

Option 2

• C.

Option 3

• D.

None of these options are correct.

A. Option 1
• 9.

Consider the MATLAB code attached. Someone thinks it contains a bug, because it produces strange results. Where is the bug?

• A.

Line 1 - The variable u is undefined.

• B.

Line 8 - The command "total(j) = 0" is on the wrong line.

• C.

Line 14 - The command "result(i,:) = total(:)" is not correct.

• D.

There is no bug in this code.

D. There is no bug in this code.
Explanation
The given code does not contain any bugs.

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