Numerical Methods MCQs Question and Answer in Engineering Maths

41

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?

A) 1.33 B) 1.67 C) 2.00 D) 2.33

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Answer: Option C

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42

What is the value of (1525)^0.2 to 2 decimal places?

A) 4.33 B) 4.36 C) 4.38 D) 4.30

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Answer: Option A

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43

Numerical integration using trapezoidal rule gives the best result for a single variable function, which is

A) linear B) parabolic C) logarithmic D) hyperbolic

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Answer: Option A

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44

Solution of the variables x₁ and x₂ for the following equations is to be obtained by employing the Newton-Raphson iterative method
equation (i) 10x₂ sin x₁ - 0.8 = 0
equation (ii) 10$${\text{x}}_2^2$$ - 10x₂ cos x₁ - 0.6 = 0
Assuming the initial values x₁ = 0.0 and x₂ = 1.0, the Jacobian matrix is

A) \[\left[ {\begin{array}{*{20}{c}} {10}&{ - 0.8} \\ 0&{ - 0.6} \end{array}} \right]\] B) \[\left[ {\begin{array}{*{20}{c}} {10}&0 \\ 0&{10} \end{array}} \right]\] C) \[\left[ {\begin{array}{*{20}{c}} 0&{ - 0.8} \\ {10}&{ - 0.6} \end{array}} \right]\] D) \[\left[ {\begin{array}{*{20}{c}} {10}&0 \\ {10}&{ - 10} \end{array}} \right]\]

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Answer: Option B

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45

The root of the function f(x) = x³ + x - 1 obtained after first iteration on application of Newton Raphson scheme using an initial guess of x₀ = 1 is

A) 0.682 B) 0.686 C) 0.750 D) 1.000

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Answer: Option C

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46

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

A) 0.01 m/s B) 0.1 m/s C) 0.2 m/s D) 1 m/s

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Answer: Option A

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47

Let x² - 117 = 0. The iterative steps for the solution using Newton-Raphson's method is given by

A) $${{\text{x}}_{{\text{k}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{k}}} + \frac{{117}}{{{{\text{x}}_{\text{k}}}}}} \right)$$ B) $${{\text{x}}_{{\text{k}} + 1}} = {{\text{x}}_{\text{k}}} - \frac{{117}}{{{{\text{x}}_{\text{k}}}}}$$ C) $${{\text{x}}_{{\text{k}} + 1}} = {{\text{x}}_{\text{k}}} - \frac{{{{\text{x}}_{\text{k}}}}}{{117}}$$ D) $${{\text{x}}_{{\text{k}} + 1}} = {{\text{x}}_{\text{k}}} - \frac{1}{2}\left( {{{\text{x}}_{\text{k}}} + \frac{{117}}{{{{\text{x}}_{\text{k}}}}}} \right)$$

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Answer: Option A

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48

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

A) 1.3, 0.6 and 0.6 respectively B) 0.6, 0.6 and 1.3 respectively C) 1.3, 1.3 and 0.6 respectively D) 1.3, 0.6 and 1.3 respectively

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Answer: Option D

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49

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?

A) 8.983 B) 9.003 C) 9.017 D) 9.045

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Answer: Option D

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50

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

A) 1.000 B) 1.098 C) 1.111 D) 1.120

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Answer: Option C

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