Prof. Dr. Fazal Rehman

Question 1: \[ \text{Evaluate } \lim_{x \to 0} \frac{\sin(x) – x}{x^3}. \] \[ \text{(a) } -\frac{1}{6}, \quad \text{(b) } 0, \quad \text{(c) } \frac{1}{6}, \quad \text{(d) } \frac{1}{2} \] Answer: A Question 2: \[ \text{Evaluate } \int_0^\infty \frac{x^2}{e^x – 1} dx. \] \[ \text{(a) } \frac{\pi^2}{6}, \quad \text{(b) } \frac{\pi^2}{3}, \quad \text{(c) } \frac{\pi^2}{2}, \quad … Read more

\[ \text{Question 1:} \] \[ \text{Which of the following methods is not used to solve nonlinear equations in one variable?} \] \[ \text{(a) Bisection Method, \quad (b) Newton-Raphson Method, \quad (c) Gaussian Elimination, \quad (d) Secant Method} \] \[ \text{Answer: C} \] \[ \text{Question 2:} \] \[ \text{The convergence rate of the Newton-Raphson method is:} … Read more

\[ \text{Question 1: If } f(x) = e^x + \ln(x), \text{ find the second derivative } f”(x). \] \[ \text{(a) } e^x – \frac{1}{x^2}, \quad \text{(b) } e^x + \frac{1}{x}, \quad \text{(c) } e^x – \frac{1}{x}, \quad \text{(d) } e^x + \frac{1}{x^2} \] \[ \text{Answer: A} \] \[ \text{Question 2: The condition number of a … Read more

Question # 1 (Covering: The determinant of an identity matrix is always 1.) \[ \textbf{Question: Calculate the determinant of the identity matrix } I \textbf{ of order } 3 \textbf{ (3×3 matrix).} \] \[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} … Read more

[latex] \[ \textbf{Q#1: Compute the Kronecker Product of the following matrices:} \] \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix} \] \[ \textbf{Step 1: Use the formula for the Kronecker product.} \] \[ A \otimes B = \begin{pmatrix} a_{11}B … Read more

Q1: Find the Conjugate of Matrix A, where: A = [(2 + 3i, 4 – i), (5 + 2i, 1 – 4i)] Q2: Find the Conjugate of Matrix B, where: B = [(3 + 4i, 2 – 3i), (7 – i, 6 + 2i)] Q3: Determine if Matrix A is Hermitian, where: A = [(3 … Read more

\[ \textbf{Exercise: Transpose of a Matrix} \] \[ \textbf{Q1: Find the Transpose of Matrix A, where:} \] \[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \] \[ \textbf{Solution:} \] \[ A^T = \text{Transpose of } A = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 … Read more

\[ \textbf{Q1: Multiply the Scalar 3 by the Matrix A, where:} \] \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \] \[ \textbf{Solution:} \] \[ 3 \cdot A = 3 \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \] \[ = \begin{pmatrix} 3 \cdot 1 & 3 \cdot 2 … Read more

[latex] \[ \textbf{Exercise: Multiplication of Matrices} \] \[ \textbf{Q1: Multiply the Square Matrices A and B, where:} \] \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \] Solution \[ \textbf{Q2: Multiply the Diagonal Matrices A and B, where:} \] … Read more

\[ \textbf{Exercise on Division of Matrices with Step-by-Step Solutions} \] \[ \textbf{Divide the matrix } A = \begin{pmatrix} 6 & 3 \\ 9 & 12 \end{pmatrix} \textbf{ by the matrix } B = \begin{pmatrix} 3 & 1 \\ 3 & 4 \end{pmatrix}. \] Solution \[ \textbf{ Divide the Square Matrix A by the Square Matrix … Read more