110年 - 110 國立政治大學_碩士班暨碩士在職專班招生考試_統計所:基礎數學#105296

1. Which of the following statements are true? (A) f is differentiable at zo implies that f is continuous at zo. (B) f(x) = x⁴ + 3 has an inflection point. (C)=1. (D) =0. (E) None of the above.

2. Let f(r) = In(1 + z). Which of the following statements are true? (A) The Taylor series about o = O js x-x² + x³-x⁴+ .... for |x |＜1. (B) The Taylor series about t = O is x -x²/2 + x³/3-x⁴/4+..., for |x |＜1. (C) f'(x) is increasing in z for r ＞ -1. (D) f"(x) is increasing in a for x > -1. (E) None of the above.

3. Consider f(x) =1/ (1 +x). Which of the following statements are true? (A) The Taylor series about x = O is 1 +x +x² +x³ +..., for  |x | ＜1. (B) The 'Taylor series aboutx = O is 1 -x+x²-x³+⋯...for |x | ＜1. (C) (D) (E) None of the above. 

4. Which of the following statements are true? (A)(B)(C)(D)(E)

5. column vector. Which of the following statements are true? (A) The gradient of f with respect to a is (2x₁, 2x₂, 2x₃)^T. (B) The Hessian of f with respect to ar is (2,2,2)^T. (C) The minimum value of f subject to 2x₁ + 2x₂ - 2x₃ = 7 is 5. (D) The maximum value of f subject to 2x₁ + 2x₂ - x₃ = 7 is 10. (E) None of the above.

6. Let f(x) = (sina)/x. Which of the following statements are true? (A) (B)(C)(D) max_xf(x)=1. (E) None of the above.

7.Let Which of the following statements are true? (A)(B) (C)(D) (E) None of the above.

8. Let A be an m x n matrix. Which of the following statements are true? (A) nullity( A ) + rank( A )=n. (B) rank( A )=rank(A^T). (C) If m = 7 and n = 5, then rank( A ) is at most 5. (D) Suppose that m = n. Then, "A is singular" "A has rank n". (E) None of the above.

9. Let A be an n x n matrix whose (i，j) component is aij. The trace of A is defined as trA = Let B and C be n x n matrices. Which of the following statements are true? (A) tr(AB)= tr(BA). (B) tr(ABC)=tr(CBA). (C) tr(A^TB)=tr(AB^T). (D) tr(A+B)=(trB)(trA). (E) None of the above. 

10. Let A be an n x n matrix whose (i, j) component is aij. Let f be a real-valued function defined on A. Let  ∇_Af( A ) be the gradient of f( A ) with respect to A； ∇_Af( A )is delined as an n x n matrix whose (i, j) entry is . Let B and C be n x n matrices. Which of the following statements are true? (A) If f( A ) = tr(AB), then ∇_Af( A )= B. (B) If f( A ) = tr(AB), then ∇_Af( A ) = B^T. (C) If f( A ) = tr(AA^TC), then ∇_Af( A ) = CA + C^TA^T. (D) If f( A ) == tr(AA^TC), then ∇_Af(  A  ) = CA + CA^T. (E) None of the above.

11. Which of the following statements are true? (A) is singular if and only if Ax = b has infinitely many solutions for every n x 1 vector b. (B) Every matrix transformnation is a linear transformation. (C) If A is a nonsingular upper triangular matrix, then the adjoint matrix is lower triangular. (D) The dimension of the zero space {0} is 0. (E) None of the above.

12. Let A be a 2 x 2 matix defined s.Which of the tolowing statements are true? (A) The characteristic function is (B) The eigenvalues are (C) 'The eigenvectors are (1,i) and (1, -i). (D) The eigenvectors are (-1,i) and (-1, -i). (E) None of the above.

13. Which of the following are projection matrices? (A)(B) (C) (D) (E) None of the above.

14. Find the matrix for the refection of R² through the a = y line. (A) (B) (C)(D) (E) None of the above.

15. Let A be an n x n matrix. Which of the following statements are true? (A) A symmetric matix A is positive definite if and only if all the eigenvalues of A are positive. (B) A is nonsingular if and only if detA=0. (C) A is singular if and only if 0 is an eigenvalue of A. (D) detA = λ₁+ λ₂ +...+  λn , where λi's are eigenvalues of A. (E) None of the above.

Part II: short answer questions
 1. (10 points) Find the limit.

2. (10 points) Evaluate the integrai.

(B) Find the value of