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

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

Please use your student number in both fields below (I. E. First name AND Last name = Student number).
Have fun - this is just to give you an idea about what to expect in your REAL exam.


Questions and Answers
  • 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

    Correct Answer(s)
    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]

    Correct Answer(s)
    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.

    Correct Answer
    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]

    Correct Answer(s)
    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

    Correct Answer
    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. 

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

    • A.

      Option 1

    • B.

      Option 2

    • C.

      Option 3

    • D.

      Option 4

    Correct Answer
    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

    Correct Answer
    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.

    Correct Answer
    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.

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

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