MCQs – Solutions of Equations in One Variable

By: Prof. Dr. Fazal Rehman | Last updated: February 8, 2025

\[ \text{Question 1:} \] \[ \text{Which of the following methods is not used to solve nonlinear equations in one variable?} \] \[ \text{(a) Bisection Method, \quad (b) Newton-Raphson Method, \quad (c) Gaussian Elimination, \quad (d) Secant Method} \] \[ \text{Answer: C} \] \[ \text{Question 2:} \] \[ \text{The convergence rate of the Newton-Raphson method is:} \] \[ \text{(a) Linear, \quad (b) Quadratic, \quad (c) Cubic, \quad (d) Logarithmic} \] \[ \text{Answer: B} \] \[ \text{Question 3:} \] \[ \text{Which of the following methods requires two initial guesses?} \] \[ \text{(a) Newton-Raphson Method, \quad (b) Secant Method, \quad (c) Fixed-Point Iteration, \quad (d) Bisection Method} \] \[ \text{Answer: B} \] \[ \text{Question 4:} \] \[ \text{Which numerical method guarantees convergence for solving a single-variable equation if the function is continuous?} \] \[ \text{(a) Newton-Raphson Method, \quad (b) Secant Method, \quad (c) Bisection Method, \quad (d) Fixed-Point Iteration} \] \[ \text{Answer: C} \] \[ \text{Question 5:} \] \[ \text{If the derivative of a function is zero at a certain point, Newton-Raphson method may:} \] \[ \text{(a) Converge faster, \quad (b) Fail to converge, \quad (c) Have cubic convergence, \quad (d) Become more stable} \] \[ \text{Answer: B} \] \[ \text{Question 6:} \] \[ \text{The Fixed-Point Iteration method is also known as:} \] \[ \text{(a) Successive Approximations, \quad (b) Chord Method, \quad (c) Tangent Method, \quad (d) False Position Method} \] \[ \text{Answer: A} \] \[ \text{Question 7:} \] \[ \text{Which method uses both function values and derivative values to approximate roots?} \] \[ \text{(a) Bisection Method, \quad (b) Newton-Raphson Method, \quad (c) Secant Method, \quad (d) Regula Falsi Method} \] \[ \text{Answer: B} \] \[ \text{Question 8:} \] \[ \text{Which method can be considered as an improvement over the Secant Method?} \] \[ \text{(a) Fixed-Point Iteration, \quad (b) Bisection Method, \quad (c) Newton-Raphson Method, \quad (d) Regula Falsi Method} \] \[ \text{Answer: C} \] \[ \text{Question 9:} \] \[ \text{In Regula Falsi method, the error reduces approximately at a rate of:} \] \[ \text{(a) Linear, \quad (b) Quadratic, \quad (c) Superlinear, \quad (d) Exponential} \] \[ \text{Answer: A} \] \[ \text{Question 10:} \] \[ \text{Which of the following methods is best suited for solving a highly oscillatory function?} \] \[ \text{(a) Newton-Raphson Method, \quad (b) Bisection Method, \quad (c) Secant Method, \quad (d) Regula Falsi Method} \] \[ \text{Answer: B} \]

SET 2

\[ \text{Question 1:} \] \[ \text{If } f(x) = x^3 – 4x + 1 \text{ has a root in the interval } (0,2), \text{ then which numerical method guarantees convergence to the root?} \] \[ \text{(a) Newton-Raphson method, \quad (b) Secant method, \quad (c) Bisection method, \quad (d) Fixed-point iteration} \] \[ \text{Answer: C} \] \[ \text{Question 2:} \] \[ \text{Consider the equation } x^5 – 3x + 1 = 0. \text{ If the root is approximated using Newton’s method, which of the following is the correct iterative formula?} \] \[ \text{(a) } x_{n+1} = x_n – \frac{x_n^5 – 3x_n + 1}{5x_n^4 – 3}, \quad \text{(b) } x_{n+1} = x_n – \frac{x_n^5 – 3x_n + 1}{5x_n^3 – 3}, \] \[ \text{(c) } x_{n+1} = x_n – \frac{x_n^5 – 3x_n + 1}{x_n^4 – 3}, \quad \text{(d) } x_{n+1} = x_n – \frac{x_n^5 – 3x_n + 1}{x_n^3 – 3} \] \[ \text{Answer: A} \] \[ \text{Question 3:} \] \[ \text{The equation } e^x – 3x = 0 \text{ has a root in } (0,1). \text{ Which numerical method is least effective in solving it?} \] \[ \text{(a) Bisection method, \quad (b) Regula Falsi method, \quad (c) Newton-Raphson method, \quad (d) Fixed-point iteration} \] \[ \text{Answer: D} \] \[ \text{Question 4:} \] \[ \text{The convergence of Newton’s method depends on the derivative of the function. If the initial guess is too close to a point where } f'(x) = 0, \text{ what can happen?} \] \[ \text{(a) The method always converges, \quad (b) The method may fail to converge, \quad (c) The method finds a different root, \quad (d) The method speeds up} \] \[ \text{Answer: B} \] \[ \text{Question 5:} \] \[ \text{Given the equation } x^3 + 4x – 1 = 0, \text{ which of the following intervals contains at least one root?} \] \[ \text{(a) } (-1,0), \quad \text{(b) } (0,1), \quad \text{(c) } (1,2), \quad \text{(d) } (-2,-1) \] \[ \text{Answer: A} \]