Exam Questions

Part I: True or False

  • Question 1: limx→3+3−x=0lim_{x \to 3^+} \sqrt{3-x} = 0 is presented as TRUE/FALSE (1 pt).
  • Question 2: If ff is a polynomial function, then limx→mf(x)=f(m)lim_{x \to m} f(x) = f(m). (1 pt)
  • Question 3: The horizontal line y=Ly = L is a horizontal asymptote to the graph of ff if and only if limx→∞f(x)=Llim_{x \to \infty} f(x) = L. (1 pt)
  • Question 4: Two matrices are equal if they have the same size and corresponding entries are identical. (1 pt)
  • Question 5: For any square matrix AA, AA is skew-symmetric if and only if A=AtA = A^t. (1 pt)
  • Question 6: The determinant of a matrix and its transpose are equal. (1 pt)

Part II: Fill in the Blanks

  • Question 1: Given f(x)={2x2−5x−3x−3,x≠3 4k,x=3f(x) = \begin{cases} \frac{2x^2 - 5x - 3}{x-3}, & x \neq 3 \ 4k, & x = 3 \end{cases} and ff is continuous at x=3x = 3, find the value of kk. (2 pts)
  • Question 2: The value of limx→1−ln(x)x−1lim_{x \to 1^-} \frac{ln(x)}{x-1} is (2 pts).
  • Question 3: If AA is an invertible square matrix and det(A)=5det(A) = 5, then find:
    • a) det(AT)=det(A^T) =
    • b) det(A−1)=det(A^{-1}) =
  • Question 4: If A=[−3]A = [-3], then ∣−3∣=|-3| =. (2 pts)
  • Question 5: If AA is a square matrix of order 3 and det(A)=2det(A) = 2, then det(3A)=det(3A) =. (2 pts)
  • Question 6: Let A=[sinecose cosesine]A = \begin{bmatrix} sine & cose \ cose & sine \end{bmatrix}, then ∣A∣=|A| =. (2 pts)