Numerical Methods - Engineering Mathematics - Metallurgical Engineering Previous Year Questions

1

2007 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2007

The probability distribution function, \( p(x) \), for a random variable, \( x \), is given by: \( p(x) = \frac{1}{\sqrt{\pi}} \exp(-x^2) \). The probability that \( x \) lies between \( x_1 = 0.6 \) and \( x_2 = 0.8 \) is [Use single-step trapezoidal rule]

A

0

B

0.069

C

0.138

D

0.560

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2

2008 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2008

The value of \(dy/dx\) for the following data set at \(x = 3.5\), computed by central difference method, is

x	1	2	3	4	5
y	0	3	8	15	24

A

3.5

B

7

C

10.5

D

14

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3

2009 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2009

What is the magnitude of the following integral using single step application of trapezoidal rule ? \[ \int_{0}^{2} (3x^2 + 4x - 2) dx \]

A

9

B

16

C

18

D

36

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4

2010 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2010

Which of the following is an iterative technique to solve a linear system of equations?

A

Gaussian elimination

B

LU decomposition

C

Newton-Raphson

D

Jacobi method

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5

2010 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2010

Given the polynomial x³ − 3x² + 4x − 2.5 = 0
Starting from a guess value x = 0 what will be the value of x after iterating twice using the Newton-Raphson method.

A

0.625

B

1.278

C

1.441

D

1.562

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6

2010 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2010

5 skilled workers can build a wall in 20 days; 8 semi-skilled workers can build a wall in 25 days; 10 unskilled workers can build a wall in 30 days. If a team has 2 skilled, 6 semi-skilled and 5 unskilled workers, how long will it take to build the wall?

A

20 days

B

18 days

C

16 days

D

15 days

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7

2011 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2011

Which one of the following methods is NOT used for numerically solving an ordinary differential equation (ODE)?

A

Euler’s method

B

Runge-Kutta method

C

Adam-Bashforth method

D

Newton-Raphson method

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8

2012 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2012

Which one of the following methods is NOT used for numerical integration?

A

Rectangular rule

B

Trapezoidal rule

C

Simpson’s rule

D

Cramer’s rule

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9

2012 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2012

If (1.001)¹²⁵⁹ = 3.52 and (1.001)²⁰⁶² = 7.85, then (1.001)³³²¹ =

A

2.23

B

4.33

C

11.37

D

27.64

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10

2013 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2013

Applying the secant method, the first approximation to the root of \(f(x) = 1 + \ln x + \frac{x}{2}\) starting with function values at \(x = 0.3\) and \(x = 0.4\), is __________

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11

2014 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2014

In the trapezoidal rule for numerical integration of a function, the nature of approximation used for the function in each interval is

A

constant

B

linear

C

parabolic

D

cubic

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12

2017 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2017

The definite integral ∫₀¹ e^−x² dx is to be evaluated numerically. Divide the integration interval into exactly 2 subintervals of equal length. Applying the trapezoidal rule, the approximate value of the integral is __________ (answer up to two decimal places)

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13

2017 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2017

Using the bisection method, the root of the equation \(x^3 + x - 1 = 0\) after three iterations is ______________ (answer up to two decimal places) (Assume starting values of x = -1 and +1)

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14

2018 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2018

Using the trapezoidal rule with two equal intervals (\(n = 2, \Delta x = 1\)), the definite integral \(\int_2^4 \frac{1}{\ln(x)} dx\) = ______ (to two decimal places).

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15

2019 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2019

The estimated value of the cube root of 37 (rounded off to two decimal places) obtained from the Newton-Raphson method after two iterations (\(x_2\)) is ______.
[Start with an initial guess value of \(x_0 = 1\)].

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16

2020 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2020

The solution (using trapezoidal rule) of the integral \( \int_{0}^{1} e^{-x} dx \) by dividing the range 0 to 1 into two equal intervals is ________ (round off to two decimal places).

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17

2021 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2021

Consider the function \( f(x) = x - \cos x \). Using Newton-Raphson method, the estimated root of \( f(x) \) after the first iteration is: ______ (round off to 3 decimal places).
Assume: Initial guess of the root = 0.5 radians.

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18

2022 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2022

Which one of the following equations will fail to converge to a root with an initial guess value of \(x = 0.5\), using the Newton-Raphson method?

A

\(x(1-x) = 0\)

B

\(e^x - 3x^2 = 0\)

C

\(x - \ln(3x) = 0\)

D

\(\tan(x) - x = 0\)

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19

2022 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2022

The integral of the function \( f(x) = 0.2 + 10x^2 \) estimated by the trapezoidal rule with a single segment from \( x = 0 \) to \( x = 1 \) is ______ (round off to 1 decimal place).

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20

2024 · Metallurgical Engineering · Engineering Mathematics · Numerical Methods

Metallurgical Engineering (MT) 2024

The following data is obtained from an experiment:

x	1	2	3
y	8	15	19

If the data is fit using the straight line \( y = mx + c \) (where \( m \) and \( c \) are constants) using the least-squares method, then the value of \( m \) is _____.
(Round off to one decimal place).

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