\[
\text{Question 1: If } f(x) = e^x + \ln(x), \text{ find the second derivative } f”(x).
\]
\[
\text{(a) } e^x – \frac{1}{x^2}, \quad \text{(b) } e^x + \frac{1}{x}, \quad \text{(c) } e^x – \frac{1}{x}, \quad \text{(d) } e^x + \frac{1}{x^2}
\]
\[
\text{Answer: A}
\]
\[
\text{Question 2: The condition number of a function } f(x) \text{ measures:}
\]
\[
\text{(a) } The numerical stability of the function, \quad \text{(b) } The accuracy of numerical differentiation,
\]
\[
\text{(c) } The error in function approximation, \quad \text{(d) } The rate of convergence of Newton’s method
\]
\[
\text{Answer: A}
\]
\[
\text{Question 3: Given the function } f(x) = x^3 – 6x^2 + 11x – 6, \text{ one of the roots is:}
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } 3, \quad \text{(d) } 4
\]
\[
\text{Answer: C}
\]
\[
\text{Question 4: The order of accuracy of the central difference approximation for the first derivative is:}
\]
\[
\text{(a) } O(h), \quad \text{(b) } O(h^2), \quad \text{(c) } O(h^3), \quad \text{(d) } O(h^4)
\]
\[
\text{Answer: B}
\]
\[
\text{Question 5: If } x \text{ is represented in floating-point as } x = 0.1011_2 \times 2^3, \text{ what is its decimal value?}
\]
\[
\text{(a) } 5.375, \quad \text{(b) } 6.25, \quad \text{(c) } 4.75, \quad \text{(d) } 7.125
\]
\[
\text{Answer: A}
\]
\[
\text{Question 6: In numerical differentiation, the truncation error in the forward difference formula is:}
\]
\[
\text{(a) } O(h), \quad \text{(b) } O(h^2), \quad \text{(c) } O(h^3), \quad \text{(d) } O(h^4)
\]
\[
\text{Answer: A}
\]
\[
\text{Question 7: The Taylor series expansion of } e^x \text{ about } x = 0 \text{ up to the fourth term is:}
\]
\[
\text{(a) } 1 + x + \frac{x^2}{2} + \frac{x^3}{6}, \quad \text{(b) } 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!},
\]
\[
\text{(c) } 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}, \quad \text{(d) } 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{5!}
\]
\[
\text{Answer: C}
\]
\[
\text{Question 8: The floating-point representation of } 1/3 \text{ leads to which type of error?}
\]
\[
\text{(a) } Round-off error, \quad \text{(b) } Truncation error, \quad \text{(c) } Absolute error, \quad \text{(d) } Relative error
\]
\[
\text{Answer: A}
\]
\[
\text{Question 9: Which of the following methods is best for solving ill-conditioned linear systems?}
\]
\[
\text{(a) } Gauss-Seidel Method, \quad \text{(b) } Jacobi Iteration, \quad \text{(c) } LU Decomposition, \quad \text{(d) } Singular Value Decomposition (SVD)
\]
\[
\text{Answer: D}
\]
\[
\text{Question 10: The round-off error in numerical computations is mainly caused by:}
\]
\[
\text{(a) } Finite precision representation, \quad \text{(b) } Approximation of derivatives,
\]
\[
\text{(c) } Finite step size in numerical integration, \quad \text{(d) } Large condition number
\]
\[
\text{Answer: A}
\]