5. Consider the differential equation tẋ + x = cos(ωt). Which of the following is true?
(A) the general solution is sin(ωt) + C, C is a constant.
(B) the particular (forced) solution is periodic with period ω
(C) the solution is NOT bounded when t grows to infinity
(D) If x(π/ω) = 1, then the solution is (sin(ωt) + π)/(ωt)
(E) none of the above

4. Consider the differential equation + x = cos(ωa t). Which of the following is true? (A) the general solution is sin(ω t) + C, C is a constant. (B) the particular (forced) solution is periodic with period ω (C) the solution is NOT bounded when t grows to infinity (D) If x( π/ ω) = 1, then the solution is (sin(ω t) +π) / (ω t) (E) none of the above

複選題6. Consider a system whose dynamics is governed by a linear constant-coefficient ODE. Suppose the impulse response of this system is . Which of the following is true? (A) the system is order (B) the characteristic equation of the system has a pair of complex roots (C) the unit step response has oscillatory behavior (D) the unit step response does NOT converge to a constant value (E) the unit step response is equal to /3

7. Consider the heat equation , ∀0 ＜ x ＜ 1, t ＞ 0 u(0,t) = u(1,t) = 0 ∀t ＞ 0 u(x, 0) = f(x), ∀0 ＜ x ＜ 1 (A) the heat equation is nonlinear (B) without considering the boundary condition, the general solution is (C) suppose f(x) = 7 sin(3x), then the solution is u(x,t) = (D) all of the above are true (E) none of the above is true

8. Define the del operator ∇ := (A) ∇(∇ × F) = 0, where F is a vector field with continuous first and second derivatives. (B) ∇(F × G) = G ⋅ (∇ × F) – F ⋅ (∇ × G), where F and G are two smooth vector fields (C) ∇ ⋅ (φF) = ∇φ ⋅ F + (∇ ⋅ F), where φ is a smooth scalar field and F a smooth vector field. (D) all of the above are true (E) none of the above is true

9. The Fourier transform of the function f(t) = u(t-2) is F(jω) = e＜sup＞aω (b+jc), where u(t) is an unit-step function. Then which of the following statements is correct? (A) a=-2, b=1, c=0. (B) a = 2, b =1, c=0. (C) a＞0, b＞1, c=0. (D) a＜2, b≠1, c≠0. (E) None of the above statements are correct.

10. Given that f(t) has the Fourier transform F(jω). Let f1(t) = f(-3t-6), and let the Fourier transform of f1(t) be F₁(jω) = . Then which of the following statements is correct? (A) a=3, b=3, c=2. (B) 0＜a＜1, -1＜b＜0, 0＜c≤2. (C) a = -3, b=1/3, c=3. (D) a＞0, b＞0, c＜0. (E) None of the above statements are correct.

11. A Fourier transform F( j ω) of a signal f (t) is shown in Fig. 1. The evaluation of E = dt is equal to α. Which of the following statements is correct? (A) α = 0 (B) α = 6√π (C) α = 3π (D) α = 3 (E) None of the above statements are correct. 

12. Which one of the following f(z), where z = x + jy is a complex variable, is entire? (A) f(z) = (B) f(z) = z Im z, Im z stands for the imaginary part of z (C) f(z) = x² + jy² (D)(E) None of the above statements are correct.

13. Let f(z) = cotz, and C be a closed path | z |= 4 in counterclockwise direction. The evaluation of  f(z)dz is c + jd. Then which of the following statements is correct? (A) c ＜ 0, d ≤ 0 (B) c ＞ 0, d ＞ 0 (C) c = 0, 15 ＜ d ＜ 20 (D) c = 0, 1 ＜ d ＜ 10 (E) None of the above statements are correct.

14. Let z be a complex number. Which of the following statements is correct? (A) │sin z │2is an unbounded function. (B) │cos z│2 is a bounded function. (C) log(i²) = 2logi (D) (E) None of the above statements are correct.

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