109年 - [非官方正解]台大_電機資訊學院_工程數學(C)#110409

1. (5%) Which are the possible solutions of the differential equation =y²ー9? (A) y=3 (B) y=-3 (C) (D) (E)

2. (5%) Which are the possible solutions of the differential equatio? (A) (B)(C) (D) (E)

4. (5%) Consider the equationwith the boundary conditions y(0)=0,y(π)=π.Which dr dx statements are correct? (A) The particular solution is (B) The general solution is (C) There is a unique solution to this boundary-value problem. (D) This differential equation is homogencous. (E) None of the above.

5. (5%) Which are the possible solutions of the differential equation ?  (A) y+ln|y-1|=x (B) 2y+1nly-1|=4x (C) 2y+1n|2y-1|=4x+4 (D) y²=2x (E) y²=-2x+1

6. (5%) Assume, Which statements are always true? (A) C₀=0 (B) C₁=0 (C)C₂ =0 (D)(E)

7. (5%) Consider the equation with the inial condition y(0) =1. Which statemenas about the function y(t) and its Laplace transform Y(s) are correct? (A) (B) (C) y(t)=sint-tsint (D) y(π)=0 (E) y(π)=-1

8. (5%) Which are the possible solutions of the given system? (A)(B)(C)(D)

9. (5%) Consider the wave equation with initial conditions Which are the possible solutions? (A)u(x,r)=f(x-at) (B) u(x,t)=f(x+at) (C)(D) (E) None of the above

10.Ealuate the Fourier transfom (A) (B) (C) (D)(E)

11. (5%) Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system:(A) 2k+g+h -0 (B) k+2g+3h =0 (C) k+2g+h = 0 (D) 3k+g+2h = 0 

12.(5%) Let a₁=. For what value(s) ofh is b in the plane

spanned by a₁ and a₂?

(A) 0

(B) -10

(C) 16

(E) 20

(D) -17

13. (5%) Determine the rank of the matrix in the following.
  (A)0 (B) 1 (C)2 (D)3 (E) 4

14. (5%) Matrix A can be written in the form A=LU, where L is m x m lower triangular matrix I's on the diagonal and U is an m x n echelon form of A. Such a factorization is called an LU factorization. Please find the L matrix for LU factorization of the following matrix A. (A)(B)(C)(D)(E)

15. (5%) Find the determinant for the following matrix. (A)-5 (B) 5 (C)1 (D)-1 (E) 0

16. (5%) Find the dimension of the subspace H = (A) 2 (B) 3 (C) 4 (D) 1 (E)0

17.Let Let , and consider the bases for R² given by B ={b₁,b₂} and C={c₁,c₂}.Find the change-of-coordinates matrix from C to B.(A)(B)(C)(D)(E)

18. (5%) Which of the following statement is True? (A) If A is a 3 ✖ 3 matrix, then det 5A=5 det A. (B) If A³=0 ,then det A =0. (C) Ifu and v are in R² and det[ u v]=10, then the area of the triangle in the plane with verices a at 0, u, v is 10. (D) IfA isn xn and det A=2, then det A³=6 . (E) Any system ofn linear equations in n variables can be solved by Cramer's rule.

19. (5%) Find the singular values of the following matrix. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4