106年 - 台大工數C#99919

1.(10%) Given an RLC circuit. The current i(t) can be expressed bywhore Which of the following are correct? (A) i(t) approaches zero at t→∞for any positive R, L,C value, (B) Fort＞0, i(t) does not have sinusoidal component, (C) (D) For bo, i(t) is proportional toe', (E) i(π)=0.

2. (5%) Solve the differential equation below. Which of the following are the possible solutions? (A) (B)(C)(D) (E)

3. (5%) For the differential equation below.
Which of the following are correct? (A)General solution form include (x)=x•cos(x), (B) General solution form include(C) Gencral solution formn include(D)(E)

4. (5%) A heart pacemaker consists of a switch, a battery, a capacitor with constant capacitance C, and the heart as a resister with constant resistance R. When the switch is closed, the capacitor charges； when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage y(t) across the heart satisties the linear differential equation : , where t is the time variable. Assume that initially, i.e, at t = O, the voltage y(t) across the heart is equal to 4 and the constant RC =1. Which of the following statements are true? (A) For t≥0, the particular solution of the lincar differential equation is (B) For t≥0,the particular solution of the linear differential equation is(C) For t≥0, the particular solution of the linear differential equation is, c is any constant, (D) For t≥0, the solution of the linear differential equation is(E) For t≥0,the solution of the linear diffcrential equation is

5. (5%) Consider one specially designed circuit with one capacitor, one inductor, and four resisters. The output voltage of the capacitor is denoted as x(t) and the output current of the inductor is denoted as y(t). Assume that initially x(0)-0 and y(0)=0 and these two devices might interact with each other by the following behavior. For the capacitor, the changing rate of x(t) is declined at a rate of -3x(), and simultaneously increased at a rate of y(t), and also positively depends on the independent current source g(t) =3t. Similarly, for the inductor；, the changing rate of y(t) declines at a rate of -4y(t), and simultaneously increases at a rate of 2x(), and also positively depends on another independent voltage source. Which of the following statements are true?  (A) (B)(C)(D)(E)

6. (5%) A forced, undamped, and resonant motion of a mass on a spring can be described in the following equation: , where x(t) is the location of the mass. In the beginning, the mass is  located at x(0)=0 and the initial velocity of the mass is equal to I, ie.,， The Laplace transformX(s) of x(t) is: (A)(B)(C) (D)(E)

7. (5%) A semi-infinite plate coincides with the region defined by. The left end is held at temperature: exp(-y), and the right end is held at temperature zero for y 2 0. The bottom of the plate is insulated. (A)g(a)=1+α,(B)f(x)=αx,(C)f(x)=α(ε-x),(D) p(y) = sin(αy), (E) p(y)=cos(ay).

8. (5%) A string is stretched and secured on the x-axis at x = 0 and x = 1 for t＞ 0, that is initially held at these points 0.01sin(3tx) and then simultaneously reicased at all points at time t = 0. The string is released from rest from the initial displacement. (Wave equation constant: a)u(x,t) = 0.01f(x)g(t) (A) g(t)=cos(πat),(B) f(x)=sin(3πx),(C) f(x)=sin(3παx), (D) g(t) = cos(3πat), (E) g(t) = sin(3πat).

9. (5%) An undamped string/mass system, in which the mass m = 1 slug and the spring constant k = 10 lb/t, is driven by the 2-periodic external force F(t) = 1-t,0 ＜t＜2； f(t+2) = f(t). Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is odd. Find a particular solution Xp(t).
(A) p(t)=nπt, (B)s=nπ, (C) k=nπ, (D) k=n²n², (E) s=n²n²,

11. (5%) Please pick the statements below that are true. (A) If A and B are m x n matrices and C is an n X p matrix, then (A + B)C = AC + BC. (B) For any matrices A and B, if A is the inverse of B^T, then A is the transpose of B-I. (C) An n x n matrix is invertible if and only if its rows are linearly independent. (D) The codomain of any function is contained in its range. (E) for every standard vector e；, then f(v) mg(v) for every v in Rⁿ.

13. (5%) Consider the matrix Which of the following values of c will make the matrix A invertible? (A) 3； (B) 4； (C) 5； (D) 6； (E) None of the above.

14. (5%) Which of the following subsets of Rⁿis a subspace? (A) (B) (C) (D) (E)

15. (5%) Which of the following statements are correct? (A) A basis for a subspace is a linearly independent subset of the subspace that is as large as possible. (B) If V is a subspace of dimension k, then every generating set for V contains exactly k vectors. (C) is any basis for Rⁿ, then there exists a vector v so that v can not be expressed as a linear combination of bi, ba...., bn. (D) Let T be a linear operator on Rⁿ, and u and V be two bases forRⁿ. Denote [T]_Bas the representative matrix of T with respect to any basis B for Rⁿ. Then, [T]u and [T]v are similar. (E) If B is a basis for Rⁿ and T is the identity operator on R", then [T]_B = I_n.

16. (5%) Which of the following statements are correct? (A) If λ is an eigenvalue of an n x n matrix A, then for any (B) If there exists some such that Av =λv, then λ is an eigenvalue of A. (C) If two matrices have the same characteristic polynomials, then they have the same eigen- values. (D) A diagonal matrix is always diagonalizable. (E) Let A and P be n X n matrices. If the columins of P form a set of n linearly independent eigenvectors of A, then AP is a diagonal matrix.