Numerical Methods MCQs Question and Answer in Engineering Maths

Newton-Raphson method is used to compute a root of the equation x² - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is

Identify the Newton-Raphson iteration scheme for finding the square root of 2.

If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?

Consider p(s) = s³ + a₂s² + a₁s + a₀ with all real coefficients. It is known that it is derivative p'(s) has no real roots. The number of real roots of p(s) is

The differential equation $$\frac{{{\text{dx}}}}{{{\text{dt}}}} = \left[ {\frac{{1 - {\text{x}}}}{\tau }} \right]$$   is discretised using Euler's numerical integration method with a time step ΔT > 0. What is the maximum permissible value of ΔT to ensure stability of the solution of the corresponding discrete time equation?

Match the correct pairs

Numerical Integration Scheme	Order of Fitting Polynomial
P. Simpson's 3/8 Rule	1. First
Q. Trapezoidal Rule	2. Second
R. Simpson's 1/3 Rule	3. Third

The minimum number of equal length subintervals needed to approximate $$\int\limits_1^2 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx}}} $$  to an accuracy of at least $$\frac{1}{3} \times {10^{ - 6}}$$  using the trapezoidal rule is

Consider the series $${{\text{x}}_{{\text{n}} + 1}} = \frac{{{{\text{x}}_{\text{n}}}}}{2} + \frac{9}{{8{{\text{x}}_{\text{n}}}}},\,{{\text{x}}_0} = 0.5$$       obtained from the Newton-Raphson method. The series converges to

The value of the function f(x) is given at n distinct values of x and its value is to be interpolated at the point x^⋆, using all the n points. The estimate is obtained first by the Lagrange polynomial, denoted by $${I_{\text{L}}}$$ and then by the Newton polynomial, denoted by $${I_{\text{N}}}$$. Which one of the following statements is correct?

The extremum (minimum or maximum) point of a function f(x) is to be determined by solving $$\frac{{{\text{df}}\left( {\text{x}} \right)}}{{{\text{dx}}}} = 0$$   using the Newton-Raphson method. Let f(x) = x³ - 6x and x₀ = 1 be the initial guess of x. The value of x after two iterations (x₂) is