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

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

None of these options are correct.

Option 2

Option 3

Option 4

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Explanation

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2) 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

15

2

5

3

0

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.

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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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.

4

5

-3

2

None of these options are correct.

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Explanation

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4) 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)

T(x) = 0

T(x) = 0.5

T(x) = x

None of the above options

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.

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.

5) Find the fourier series representation of the function shown.

Option 1

Option 2

Option 3

None of these options are correct.

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Explanation

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

Line 1 - The variable u is undefined.

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

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

There is no bug in this code.

The given code does not contain any bugs.

Explanation

The given code does not contain any bugs.

7) For the ODE shown, what is the correct Finite Difference (FD) discretization?

Option 1

Option 2

Option 3

Option 4

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Explanation

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

[-4; 1]

[1; 1]

[2; 2]

[4; -1]

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].

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

[1; -2]

[2; -1]

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

[-2; 1]

[1; 2]

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.

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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10) Match the following signals with their fourier series equivalent.

Fourier Series 1

Fourier Series 2

Fourier Series 3

Fourier Series 4

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