108年 - [非官方正解]108 國立臺灣大學_碩士班招生考試：工程數學(C)#124656

1. (5%) Which of the following is the possible integrating factor for y' -y =x⁴?
(A) e^-2
(B) x^-2
(C) ln(^x-2)
(D) e^2x-2
(E) e^-2x

2. (5%) Find the inverse Laplace transform of the function F(s) =.
(A) cos3x + sin3x
(B) cosh3x + sinh3x
(C) e^3x cos2x + e^3xsin2x
(D) e^3x cosh2x + e^3xsinh2x
(E) e^3x + e^-3x

3. (5%) Which are the possible solutions of the differential equation y" - 4y'+ 4y = e^t?
(A) y = e^t
(B) y = e^2t
(C) y = e^t + e^2t
(D) y = te^t + e^2t
(E) y = e^t + te^2t

4. (5%) The mass-spring system is described by x" +bx' + 4x = 0, x(0)=4, x'(0)= 0.
Under which condition, the system is in the state of underdamped motion?
(A)b＞4
(B)0＜b＜4
(C)b = 4
(D)b = 2
(E)None of above

5. (5%) Solve the equation y⁽⁵⁾ - y⁽⁴⁾ – 2y ⁽³⁾ + 2y" + y' = y. Which are the possible solutions?
(A)y = C₁e^x + c₂xe^x+C₃x²ex + C₄x³ex+c₅xe^-x
(B)y = C₁ex + C₂xe^x+C₃e^-x + C₄xe^-x+c₅x²e^-x
(C)y = C₁e^x + C₂xe^x+C₃e^-x + C4e^-2x+C₅xe-^2x
(D)y = c₁e^x + c₂xe^x+C3x²e^x + C₄e^-x+c₅xe^-x
(E) y = C₁e^x + C₂xe^x+C₃e^x + C₄e^2x+C₅e^-2x

6. (5%) Which boundary value problems have nontrivial solutions? (Trivial solution is defined as y(x) = 0.)
(A) y" + 2y' - 3y = 0； y(0) = 0, y'(1) = 0
(B) y" = 0； y(-1) = 0, y(1) – 2y'(1) = 0
(C) y" + y = 0； y(0) = 0, y(π) = 0
(D) y" + y = 0； y(0) = 0, y(π) = 1
(E) All of above

7. (5%) Which functions satisfy the wave equation?
(A)u(x,t) = (x² + k²)t²
(B) u(x, t) = sin x cos kt
(C) u(x, t) = tan⁻¹(x + kt)
(D)u(x, t) = ln(x + kt)
(E) u(x, t) = e⁻ᵏᵗ sin x

8. (5%) The initial value problem is described as follows: y" + 4y = 0； y(0) = 2, y'(0) = 2. The Laplace transform Y(s) of the solution y(t) is:
(A)
(B)
(C)
(D)
(E)

9. (5%) The differential equation x" + 3x' - 5x = sin t is equivalent to the system:
(A)x' = 3x - 5y； y' = sin t
(B) x' = 5x - 3y； y' = sin t
(C) x' = y； y' = 5x - 3y + sin t
(D)x' = y； y' = 3x - 5y + sin t
(E) None of above

10. (5%) The differential equation is y' = in the standard form, and 2xydx + x²dy = 0 in a differential form. This differential equation is
(A) linear
(B) homogenous
(C) exact
(D) separable
(E) Bernoulli

11. (10%) Which of the following are correct?
(A) The system defined by F(x,y)=(x², x) is linear.
(B) The system defined by F(x,y)=(dx/dt, x) is linear.
(C) A and B are m x n matrices. If Aw = Bw for all w in Rn, then A = B.
(D) The columns of matrix A contains zero vector. If Ax=b have solution, it will have infinite solutions.
(E) The zero vector of Rⁿ is within the span of any finite subset of Rⁿ.

12. (10%) For the matrix A, which of the following are correct?
A =
(A) The rank of A is 4.
(B) The rank of A is 3.
(C) The dimension of null space of A is 0.
(D) The dimension of null space of A is 1.
(E) The dimension of null space of A is 2.

13. (5%) Which of the following are correct?
(A) Below vector set S is dependent.
S =
(B) Invertible matrix must be square matrix.
(C) Square matrix must be invertible.
(D) The inverse of matrix

(E) A n x n matrix is invertible if its columns span Rⁿ.

14. (10%) For the matrix below, which of the following are correct?

(A) -1 is its eigenvalue.
(B) 1 is its eigenvalue.
(C) -3 is its eigenvalue.
(D) Zero vector belongs to its eigenvector.
(E) [1 0 2]^T is a basis of its eigenspace.

15. (5%) Which of the following are correct?
(A) An eigenvalue of a matrix has infinite number of eigenvectors.
(B) If det(A – λI_n) = 0, λ is the eigenvalue of n x n matrix A.
(C) If λ is the eigenvalue of matrix A, the columns of (A -λI_n) are independent.
(D) If λ is the eigenvalue of matrix A, there exist a non-zero vector v such that Av = λv.
(E) The number of the eigenvalue of n x n matrix A may larger than n.

16. (10%) Which of the following are correct?
(A) Any orthogonal set of vectors is linearly independent.
(B) Any orthonormal set of vectors is linearly independent.
(C) If n x n matrix A is orthogonal, A is invertible.
(D) If n x n matrix A is orthogonal, AA^T=I_n.
(E) If n x n matrix A is orthogonal, det( A ) = 1