Q1: Which of the following methods is primarily used for solving large systems of linear equations efficiently?
(A) Gauss-Seidel method
(B) LU decomposition
(C) Newton’s method
(D) Trapezoidal rule
Answer: (B) LU decomposition
Q2: In numerical differentiation, which of the following is the most accurate method for approximating the first derivative of a function?
(A) Forward difference method
(B) Backward difference method
(C) Central difference method
(D) Richardson extrapolation
Answer: (C) Central difference method
Q3: The Runge-Kutta method is an example of which type of numerical method?
(A) Direct method
(B) Iterative method
(C) Predictor-corrector method
(D) Finite difference method
Answer: (C) Predictor-corrector method
Q4: Which of the following is a common issue encountered in numerical methods when solving large systems of equations?
(A) Loss of precision due to rounding errors
(B) Convergence to an incorrect solution
(C) Lack of convergence
(D) All of the above
Answer: (D) All of the above
Q5: Which method is commonly used for solving ordinary differential equations (ODEs) with stiff behavior?
(A) Euler’s method
(B) Adams-Bashforth method
(C) Implicit methods
(D) Trapezoidal rule
Answer: (C) Implicit methods
Q6: In the context of numerical integration, what does the term “convergence” refer to?
(A) The process of finding an approximate solution to an equation
(B) The condition under which the solution of a numerical method approaches the exact solution as the step size decreases
(C) The stability of the algorithm as the input size increases
(D) The ability of a method to handle large matrices efficiently
Answer: (B) The condition under which the solution of a numerical method approaches the exact solution as the step size decreases
Q7: The stability of a numerical method for solving differential equations is crucial in avoiding which of the following?
(A) Inaccurate results due to rounding errors
(B) Instability leading to unbounded solutions
(C) Overfitting of the data
(D) Numerical underflow
Answer: (B) Instability leading to unbounded solutions
Q8: In the context of interpolation, which of the following is true for Lagrange interpolation?
(A) It minimizes the error between the interpolated points
(B) It uses polynomial functions to interpolate the data points
(C) It requires the use of an iterative method
(D) It can only be used for equispaced data points
Answer: (B) It uses polynomial functions to interpolate the data points
Q9: Which of the following methods is used for solving large-scale linear systems when the matrix is sparse?
(A) Gaussian elimination
(B) Conjugate gradient method
(C) LU decomposition
(D) Crank-Nicolson method
Answer: (B) Conjugate gradient method
Q10: In the context of Fourier analysis, which of the following is true about the Fast Fourier Transform (FFT)?
(A) FFT can only be applied to time-domain signals
(B) FFT is a method to compute the Discrete Fourier Transform (DFT) efficiently
(C) FFT requires continuous data inputs
(D) FFT is used to solve partial differential equations
Answer: (B) FFT is a method to compute the Discrete Fourier Transform (DFT) efficiently