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

40. 依平均地權條例規定，地價稅之累進起點地價，係為各該直轄市或縣(市)土地多少公畝之平均地價? (A)五 (B)七 (C)九 (D)十

1. When using the Gaussian elimination to solve a linear equation Ax = b, elementary row operations (or multiplication by elementary matrices) are applied to the augmented matrix [Ab]. Actually, there are many places in linear algebra where such a technique plays its role； e.g., the null space of A can be determined by setting b = 0. Which of the following cannot be determined by applying such a technique? (A) the range of A (B) the inverse of A (if it is nonsingular) (C) the QR factorization of A (D) the LU factorization of A (if it is square) (E) the determinant of A (if it is square).

2. Consider the system represented by the differential equation + kx = 0. Which of the following is true? (A) the system is critically damped when k = 1 (B) the system is underdamped when k = 1/2 (C) the system is overdamped when k = 0.3 (D) the damping ratio of the system is increased when increasing k (E) none of the above

3. Consider the differential equation + kx = cos(ω t) and its sinusoidal solution . Which of the following is true? (A) for a fixed b, the amplitude of is maximized when k = ω2 (B) for a fixed b, the maximal amplitude of is 1/b (C) for fixed b and k, the amplitude of always grows as ω increases (D) the period of is 2ω (E) the period of depends on b and k.

5. 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 system is order (C) the characteristic equation of the system has a pair of complex roots (D) one root of the characteristic equation is less than -5 (E) response to a step function does NOT converge to a constant value

6. Consider the system described in Question 5 again. The unit step response of this system (A) has oscillatory behavior (B) converges to 1 (C) converges to 1/3 (D) (E)

7. What is the solution to the ODE = 4t - 3x ? (A) x(t) = constant (B) x(t) = -ct3 (C) x(t) = + t (D) x(t) = + 4t (E) none of the above

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

9. The linear combination of a set of vectors is an essential element in linear algebra. We say a set V is invariant under linear combination if the implication " and any set of vectors the set of all linear combinations V" holds. And we say a mapping L defined on a set X is invariant under linear combination if the form of linear combination is unchanged under L, or more precisely the statement ", and any set of vectors X, the identity = holds" is true. Which one of the following statements related to linear combination is false? (A) Let S be a subset of a vector space V. Then S is a subspace of V if S is invariant under linear combination. (B) Let  be a set of k subspaces of a vector space W and denote as the set of all linear combinations of the form , with each chosen freely from . Then spa is also a subspace of W with = (C) Let A and B be two matrices and denote C := AB. Then each column of C is a linear combination of all columns of A, and so rank≤ rank is implied.  (D) A mapping L between two vector spaces is a linear transformation if and only if it is invariant under linear combination. (E) Let (V, (. ,. )V) be an inner product space. Then (. , .)V is invariant under linear combination at either one of its two arguments.

10. Let U, V, W be vector spaces and T: U →rightarrow V, S: V →rightarrow W be linear transformations. Also denote the set of all linear transformations from V to W by (V,W) (e.g. S (V,W)). Which one of the following statements is false? (A) The set ((V,W), +, ‧), where (α ‧S1 + β ‧ S2)(v) =α ‧S1(v) +β S2(v) for any v V, (V,W), and α ‧ βR, is also a vector space with dim\mathcal(V,W) = dimV + dimW. (B) The mapping ST(.) defined as ST(u) := S(T(u)) for any u \in U is also a linear transformation from U to W. (C) If both S and T are one-to-one, then their combination ST(.) is also one-to-one. (D) If S: V →rightarrow W is invertible, then the mapping from W to V is also a linear transformation. (E) Suppose that the combination ST(.) is onto. Then at least one of S and T is onto.