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With respect to the numerical evaluation of the definite integral $${\text{K}} = \int_{\text{a}}^{\text{b}} {{{\text{x}}^2}{\text{dx,}}} $$   where a and b are given, which of the following statements is/are TRUE?<br>I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.<br>II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.

Correct Answer: Both I and II

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?

Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$    where z = x + iy. The line c is shown in the figure below
Complex Variable mcq question image
The value of $$I$$ is

A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale).
Numerical Methods mcq question image
If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be

P(0, 3), Q(0.5, 4) and R(1, 5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson's rule within limits x = 0 and x = 1 for the curve. The difference between the two results will be

Consider the following statements regarding nucleate boiling (i) The temperature of the surface is greater than the saturation temperature of the liquid (ii) Bubbles are created by the expansion of entrapped gas oil vapor at small cavities in the surface (iii) The temperature is greater than that in film boiling (iv) The heat transfer from the surface to the liquid is greater than that in the film boiling Which of these statements are correct?

In Arun's opinion, his weight is greater than 65 kg but leas than 72 kg. His brother does not agree with Arun and he thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all of them are correct in their estimation, what is the average of diferent probable weights of Arun ?

In Arun's opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Arun and he thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Arun?

In Arun's opinion, his weight is greater than 65 kg but less than 72 kg. His brother doest not agree with Arun and he thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Arun?

L is older than M and taller than N. O is younger than P, older than N and shorter than Q. R is older than S, younger than N, shorter than S and taller than Q. S is older than L, M, O, P, Q, R and S is necessarily false?

In Arun’s opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Arun and he thinks that Arun’s weight is greater is that 60 kg but less than 70 kg. His mother’s view is that his weight cannot be greater than 68 kg. If all of them are correct in their estimation, what is the average of different probable weights of Arun?