Numerical Methods - Engineering Mathematics - Production & Industrial Engineering Previous Year Questions

1

2007 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2007

Matching Exercise. Choose the correct one out of the alternatives A,B,C,D

A

P-3, Q-2, R-4, S-1

B

P-2, Q-4, R-3, S-1

C

P-1, Q-2, R-3, S-4

D

P-1, Q-3, R-2, S-4

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2

2008 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2008

If the interval of integration is divided into two equal intervals of width 1.0, the value of the definite integral $\int_0^2 \log_e x \, dx$, using Simpson's one-third rule, will be

A

0.50

B

0.80

C

1.00

D

1.29

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3

2008 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2008

If the interval of integration is divided into two equal intervals of width 1.0, the value of the definite integral \(\int_{1}^{3} \log_e x \, dx\), using Simpson's one-third rule, will be

A

0.50

B

0.80

C

1.00

D

1.29

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4

2009 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2009

During the numerical solution of a first order differential equation using the Euler (also known as Euler Cauchy) method with step size \(h\), the local truncation error is of the order of

A

\(h^2\)

B

\(h^3\)

C

\(h^4\)

D

\(h^5\)

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5

2009 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2009

During the numerical solution of a first order differential equation using the Euler (also known as Euler Cauchy) method with step size h, the local truncation error is of the order of

A

\(h^2\)

B

\(h^3\)

C

\(h^4\)

D

\(h^5\)

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6

2009 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2009

The area under the curve shown, between \(x = 1\) and \(x = 3\) is to be evaluated using the trapezoidal rule. The following points on the curve are given.

Point	X coordinate(m)	Y coordinate(m)
1	1	1
2	2	4
3	3	9

The evaluated area (in \(m^2\)) will be

A

7

B

8.67

C

9

D

18

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7

2009 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2009

The area under the curve shown, between \( x = 1 \) and \( x = 3 \) is to be evaluated using the trapezoidal rule. The following points on the curve are given.

Point	X coordinate(m)	Y coordinate(m)
1	1	1
2	2	4
3	3	9

The evaluated area (in m²) will be

A

7

B

8.67

C

9

D

18

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8

2010 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2010

The following algorithm computes the integral \( J = \int_{a}^{b} f(x) dx \) from the given values \( f_j = f(x_j) \) at equidistant points: \( x_0 = a \; ; \; x_1 = x_0 + h \; ; \; ......; \; x_{2m} = x_0 + 2mh = b \)
Compute \( S_0 = f_0 + f_{2m} \)
\( S_1 = f_1 + f_3 + ...... + f_{2m-1} \)
\( S_2 = f_2 + f_4 + ...... + f_{2m-2} \)
\( J = \frac{h}{3} (S_0 + 4S_1 + 2S_2) \)
The rule of numerical integration, which uses the above algorithm, is

A

Rectangular rule

B

Trapezoidal rule

C

Four-point rule

D

Simpson's rule

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9

2010 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2010

Euler's method of integration is applied to the initial value problem: \( \frac{dy}{dx} = 2x ; \; y(0) = 0 \).
If the step size \( h = 0.2 \), then the error in computation (in percentage) after 5 steps would be

A

0

B

10

C

20

D

30

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10

2010 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2010

Euler's method of integration is applied to the initial value problem: dy/dx = 2x, y(0) = 0. If the step size h = 0.2, then the error in computation (in percentage) after 5 steps would be

A

0

B

10

C

20

D

30

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11

2011 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2011

The value of \(\int_0^1 e^{-x^2} dx\), using trapezoidal rule for 10 trapezoids, is equal to

A

0.6778

B

0.7165

C

0.6985

D

0.7462

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12

2011 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2011

The value of \[ \int_{0}^{1} e^{-x^2} dx \], using trapezoidal rule for 10 trapezoids, is equal to

A

0.6778

B

0.7165

C

0.6985

D

0.7462

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13

2014 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2014

Using the Simpson’s 1/3rd rule, the value of \( \int y dx \) computed, for the data given below, is ______.

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14

2014 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2014

If the equation \( \sin(x) = x^2 \) is solved by Newton Raphson’s method with the initial guess of x=1, then the value of x after 2 iterations would be ______.

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15

2015 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2015

In numerical integration using Simpson’s rule, the approximating function in the interval is a

A

constant

B

straight line

C

cubic B-Spline

D

parabola

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16

2016 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2016

To solve the equation \(2 \sin x = x\) by Newton-Raphson method, the initial guess was chosen to be \(x = 2.0\). Consider \(x\) in radian only. The value of \(x\) (in radian) obtained after one iteration will be closest to

A

\(-8.101\)

B

\(1.901\)

C

\(2.099\)

D

\(12.101\)

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17

2017 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2017

Using Simpson’s 1/3 rule for numerical integration, the consecutive points are joined by a

A

line

B

parabola

C

polynomial with power 3

D

polynomial with power 1/3

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18

2017 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2017

Runge-Kutta fourth order method is used to solve the differential equation \(\frac{dy}{dx} = y - x\). If the initial value \(y(0) = 2\) and step-size is 0.1, then the value of \(y(0.1)\) is _____________ (up to three decimal places).

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19

2018 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2018

In order to evaluate the integral \( \int_{0}^{1} e^x dx \) with Simpson's 1/3rd rule, values of the function \( e^x \) are used at \( x = 0.0, 0.5 \) and 1.0. The absolute value of the error of numerical integration is

A

0.000171

B

0.000440

C

0.000579

D

0.002718

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20

2019 · Production & Industrial Engineering · Engineering Mathematics · Numerical Methods

Production & Industrial Engineering (PI) 2019

The values of a function \(f(x)\) over the interval \([0,4]\) are given in the table below:

\(x\)	0	1	2	3	4
\(f(x)\)	1	0.5	0.2	0.1	0.06

Then, according to the trapezoidal rule, the value of the integral \(\int_{0}^{4} f(x)dx\) is ________ (round off to 2 decimal places).

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