106年 - 106 國立中山大學_碩士班招生考試_資工系(乙組)：工程數學#105789

試卷資訊

所屬科目：中山◆資工◆工程數學

選擇題 (0)

申論題 (14)

1.1 Show that AB is idempotent if AB = BA.

1.2 Show that if A is idempotent, then is idempotent.

1.3 Is A+B idempotent? Justify your answer.

1.4Find all values of k for which kA is also idempotent.

2.1 Sketch the periodic function x(t) over the time interval -10 s t ≤ 20.

2.2 Determine the DC coefficient of the Fourier series, ao.

2.3 Use the Fourier analysis integral to find al, the first Fourier series coefficient.

2.4 If we add a constant valie of one to x(t), we obtain the signal y(t) = 1 + x(1) with y(t) given over
one period by
The signal can be represented by a Fourier series, but with different coefficients:
Explain how bo and b! are related to ao and a . Note: you should not have to evaluate any new integrals
explicitly to answer this question.

3.1 Find the characteristic polynomial.

3.2 Find the eigenvalues,

3.3 And Find the associated eigenvectors.

4. Using Laplace transform and showing the details of your work, solve the initial value problem. y₁'=-2y₁+3y₂, y₂=4y₁-y₂, y₁(0)=4, y₂(0)=3,

5.Find the general solution of the following differential equation.

y"+4y'+4y=

sin2x

6. Solve the following initial value problem.

y'"-2y"+4y'-8y=0,y(0)=-1, y'(0)=30, y"(0)=28,