Numerical Methods MCQs Question and Answer in Engineering Maths

In numerical integration using Simpson's rule, the approximating function in interval is a

Using Simpson's 1/3 rule for numerical integration, the consecutive points are joined by a

The differential equation $$\frac{{{\text{dx}}}}{{{\text{dt}}}} = \frac{{4 - {\text{x}}}}{\tau },$$   with x(0) = 0 and the constant $$\tau $$ > 0, is to be numerically integrated using the forward Euler method with a constant integration time step T. The maximum value of T such that the numerical solution of x converges is

Match the following:

List-I	List-II
P. 2nd order differential equation	1. Runge-Kutta Method
Q. Non-linear algebraic equation	2. Newton-Raphson Method
R. Linear algebraic equation	3. Gauss Elimination
S. Numerical integration	4. Simpson's rule

Match the application to appropriate numerical method.

Application	Numerical Method
P1: Numerical integration	M1: Newton-Raphson Method
P2: Solution to a transcendental equation	M2: Runge-Kutta Method
P3: Solution to a system of linear equations	M3: Simpson's 1/3-rule
P4: Solution to a differential equation	M4: Gauss Elimination Method

The estimate of $$\int\limits_{0.5}^{1.5} {\frac{{{\text{dx}}}}{{\text{x}}}} $$  obtained using Simpson's rule with three-point function evaluation exceeds the exact value by

The integral $$\int\limits_{{{\text{x}}_1}}^{{{\text{x}}_2}} {{{\text{x}}^2}{\text{dx}}} $$  with x₂ > x₁ > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $$I$$ is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

The error in $${\left. {\frac{{\text{d}}}{{{\text{dx}}}}{\text{f}}\left( {\text{x}} \right)} \right|_{{\text{x}} = {{\text{x}}_0}}}$$   for a continuous function estimated with h = 0.03 using the central difference formula $${\left. {\frac{{\text{d}}}{{{\text{dx}}}}{\text{f}}\left( {\text{x}} \right)} \right|_{{\text{x}} = {{\text{x}}_0}}} = \frac{{{\text{f}}\left( {{{\text{x}}_0} + {\text{h}}} \right) - {\text{f}}\left( {{{\text{x}}_0} - {\text{h}}} \right)}}{{2{\text{h}}}},$$        is 2 × 10^-3. The values of x₀ and f(x₀) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately