Numerical Methods — Part 2 MCQs with Explanations

2.00 is the correct answer to “The differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = 0.25{{\text{y}}^2}$$ is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1”. This question is standard in competitive examination syllabi for Numerical Methods.

4.33 is the correct answer to “What is the value of (1525) 0.2 to 2 decimal places”. This question is standard in competitive examination syllabi for Numerical Methods.

linear is the correct answer to “Numerical integration using trapezoidal rule gives the best result for a single variable function, which is”. This question is standard in competitive examination syllabi for Numerical Methods.

\[\left[ {\begin{array}{*{20}{c}} {10}&0 \\ 0&{10} \end{array}} \right]\] is the correct answer to “Solution of the variables x 1 and x 2 for the following equations is to be obtained by employing the Newton-Raphson iterative method equation (i) 10x 2 sin x 1 - 0.8 = 0 equation (ii) 10$${\text{x}}_2^2$$ - 10x 2 cos x 1 - 0.6 = 0 Assuming the initial values x 1 = 0.0 and x 2 = 1.0, the Jacobian matrix is”. This question is standard in competitive examination syllabi for Numerical Methods.

0.750 is the correct answer to “The root of the function f(x) = x 3 + x - 1 obtained after first iteration on application of Newton Raphson scheme using an initial guess of x 0 = 1 is”. This question is standard in competitive examination syllabi for Numerical Methods.

0.01 m/s is the correct answer to “The velocity v (in m/s) of a moving mass, starting from rest, given as $$\frac{{{\text{dv}}}}{{{\text{dt}}}} = {\text{v}} + {\text{t}}{\text{.}}$$ Using Euler forward difference method (also known as Cauchy-Euler method) with a step size of 0.1 s, the velocity at 0.2 s evaluate to”. This question is standard in competitive examination syllabi for Numerical Methods.

$${{\text{x}}_{{\text{k}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{k}}} + \frac{{117}}{{{{\text{x}}_{\text{k}}}}}} \right)$$ is the correct answer to “Let x 2 - 117 = 0. The iterative steps for the solution using Newton-Raphson's method is given by”. This question is standard in competitive examination syllabi for Numerical Methods.

1.3, 0.6 and 1.3 respectively is the correct answer to “A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale). If we use the Newton-Raphson method to find the roots of f(x) = 0 using x 0 , x 1 and x 2 respectively as initial guesses, the roots obtained would be”. This question is standard in competitive examination syllabi for Numerical Methods.

9.045 is the correct answer to “Function f is known at the following points x 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 f(x) 0 0.09 0.36 0.81 1.44 2.25 3.24 4.41 5.76 7.29 9.00 The value of $$\int_0^3 {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$ computed using the continuous at x = 3”. This question is standard in competitive examination syllabi for Numerical Methods.

1.111 is the correct answer to “The integral $$\int_1^3 {\frac{1}{{\text{x}}}{\text{dx,}}} $$ when evaluated by using Simpson's 1/3 rule on two equal subintervals each of length 1, equals”. This question is standard in competitive examination syllabi for Numerical Methods.

parabola is the correct answer to “In numerical integration using Simpson's rule, the approximating function in interval is a”. This question is standard in competitive examination syllabi for Numerical Methods.

parabola is the correct answer to “Using Simpson's 1/3 rule for numerical integration, the consecutive points are joined by a”. This question is standard in competitive examination syllabi for Numerical Methods.

$$2\tau $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

P-1, Q-2, R-3, S-4 is the correct answer to “Match the following: List-I List-II P. 2 nd 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”. This question is standard in competitive examination syllabi for Numerical Methods.

P1-M3, P2-M1, P3-M4, P4-M2 is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

0.012 is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

J > $$I$$ is the correct answer to “The integral $$\int\limits_{{{\text{x}}_1}}^{{{\text{x}}_2}} {{{\text{x}}^2}{\text{dx}}} $$ with x 2 > x 1 > 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”. This question is standard in competitive examination syllabi for Numerical Methods.

9.0 × 10-4 is the correct answer to “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 x 0 and f(x 0 ) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately”. This question is standard in competitive examination syllabi for Numerical Methods.