Numerical Methods MCQs Question and Answer in Engineering Maths Page-4 section-1

31.
A calculator has accuracy up to 8 digits after decimal place. The value of $$\int\limits_0^{2\pi } {\sin {\text{x dx}}} $$ when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is

A. 0.00000

B. 1.0000

C. 0.00500

D. 0.00025

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32.
The following table lists an n^th order polynomial f(x) = a_n xⁿ + a_{n - 1} x^{n - 1} + , .... a₁x + a₀ and the forward difference evaluated at equally spaced values of x. The order of the polynomial is

x f(x) Δf Δ²f Δ³f

-0.4 1.7648 -0.2965 0.089 -0.03

-0.3 1.4683 -0.2075 0.059 -0.0228

-0,2 1.2608 -0.1485 0.0362 -0.0156

-0.1 1.1123 -0.1123 0.0206 -0.0084

0 1 -0.0917 0.0122 -0.0012

0.1 0.9083 -0.0795 0.011 0.006

0.2 0.8288 -0.0685 0.017 0.0132

A. 1

B. 2

C. 3

D. 4

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33.
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

A. 0

B. 0.25

C. 0.5

D. 1

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34.
The recursion relation to solve x = e^-x using Newton-Raphson method is

A. $${{\text{x}}_{{\text{n}} + 1}} = {{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}$$

B. $${{\text{x}}_{{\text{n}} + 1}} = {{\text{x}}_{\text{n}}} - {{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}$$

C. $${{\text{x}}_{{\text{n}} + 1}} = \left( {1 + {{\text{x}}_{\text{n}}}} \right)\frac{{{{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}}}{{1 + {{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}}}$$

D. $${{\text{x}}_{{\text{n}} + 1}} = \frac{{{\text{x}}_{\text{n}}^2 - {{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}\left( {1 + {{\text{x}}_{\text{n}}}} \right) - 1}}{{{{\text{x}}_{\text{n}}} - {{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}}}$$

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35.
Given that one root of the equation x³ - 10x² + 31x - 30 = 0 is 5, the other two roots are

A. 2 and 3

B. 2 and 4

C. 3 and 4

D. -2 and -3

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36.
The bisection method is applied to compute a zero of the function f(x) = x⁴ - x³ - x² - 4 in the interval [1, 9]. The method converges to a solution after . . . . . . . . iterations.

A. 1

B. 3

C. 5

D. 7

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37.
The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x² - N = 0. If i denotes the iteration index, the correct iterative scheme will be

A. $${{\text{x}}_{{\text{i}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{i}}} + \frac{{\text{N}}}{{{{\text{x}}_{\text{i}}}}}} \right)$$

B. $${{\text{x}}_{{\text{i}} + 1}} = \frac{1}{2}\left( {{\text{x}}_{\text{i}}^2 + \frac{{\text{N}}}{{{\text{x}}_{\text{i}}^2}}} \right)$$

C. $${{\text{x}}_{{\text{i}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{i}}} + \frac{{{{\text{N}}^2}}}{{{{\text{x}}_{\text{i}}}}}} \right)$$

D. $${{\text{x}}_{{\text{i}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{i}}} - \frac{{\text{N}}}{{{{\text{x}}_{\text{i}}}}}} \right)$$

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38.
During the numerical solution of a first order differential equation using Euler method with step size h, the local truncation error is of order of

A. h²

B. h³

C. h⁴

D. h⁵

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39.
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?
I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.

A. I only

B. II only

C. Both I and II

D. Neither I nor II

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40.
The following equation needs to be numerically solved using the Newton-Raphson method. x³ + 4x - 9 = 0. The iterative equation for this purpose is (k indicates the iteration level)

A. $${{\text{x}}_{{\text{k}} + 1}} = \frac{{2{\text{x}}_{\text{k}}^3 + 9}}{{3{\text{x}}_{\text{k}}^2 + 4}}$$

B. $${{\text{x}}_{{\text{k}} + 1}} = \frac{{3{\text{x}}_{\text{k}}^2 + 4}}{{2{\text{x}}_{\text{k}}^2 + 9}}$$

C. $${{\text{x}}_{{\text{k}} + 1}} = {{\text{x}}_{\text{k}}} - 3{\text{x}}_{\text{k}}^2 + 4$$

D. $${{\text{x}}_{{\text{k}} + 1}} = \frac{{4{\text{x}}_{\text{k}}^2 + 3}}{{9{\text{x}}_{\text{k}}^2 + 2}}$$

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x	f(x)	Δf	Δ²f	Δ³f
-0.4	1.7648	-0.2965	0.089	-0.03
-0.3	1.4683	-0.2075	0.059	-0.0228
-0,2	1.2608	-0.1485	0.0362	-0.0156
-0.1	1.1123	-0.1123	0.0206	-0.0084
0	1	-0.0917	0.0122	-0.0012
0.1	0.9083	-0.0795	0.011	0.006
0.2	0.8288	-0.0685	0.017	0.0132

Numerical Methods MCQs Question and Answer in Engineering Maths

31. A calculator has accuracy up to 8 digits after decimal place. The value of $$\int\limits_0^{2\pi } {\sin {\text{x dx}}} $$ when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is

Answer & Solution

Answer & Solution

33. 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

Answer & Solution

34. The recursion relation to solve x = e-x using Newton-Raphson method is

Answer & Solution

35. Given that one root of the equation x3 - 10x2 + 31x - 30 = 0 is 5, the other two roots are

Answer & Solution

36. The bisection method is applied to compute a zero of the function f(x) = x4 - x3 - x2 - 4 in the interval [1, 9]. The method converges to a solution after . . . . . . . . iterations.

Answer & Solution

37. The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x2 - N = 0. If i denotes the iteration index, the correct iterative scheme will be

Answer & Solution

38. During the numerical solution of a first order differential equation using Euler method with step size h, the local truncation error is of order of

Answer & Solution

Answer & Solution

40. The following equation needs to be numerically solved using the Newton-Raphson method. x3 + 4x - 9 = 0. The iterative equation for this purpose is (k indicates the iteration level)

Answer & Solution

31.
A calculator has accuracy up to 8 digits after decimal place. The value of $$\int\limits_0^{2\pi } {\sin {\text{x dx}}} $$ when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is

33.
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

34.
The recursion relation to solve x = e^-x using Newton-Raphson method is

35.
Given that one root of the equation x³ - 10x² + 31x - 30 = 0 is 5, the other two roots are

36.
The bisection method is applied to compute a zero of the function f(x) = x⁴ - x³ - x² - 4 in the interval [1, 9]. The method converges to a solution after . . . . . . . . iterations.

37.
The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x² - N = 0. If i denotes the iteration index, the correct iterative scheme will be

38.
During the numerical solution of a first order differential equation using Euler method with step size h, the local truncation error is of order of

40.
The following equation needs to be numerically solved using the Newton-Raphson method. x³ + 4x - 9 = 0. The iterative equation for this purpose is (k indicates the iteration level)