Numerical Methods - Engineering Mathematics - Textile Engineering & Fibre Science Previous Year Questions

1

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2007

The following table gives the values of a function f(x) at points x_j in an interval [0,1], where j represents the index of the point in the given interval

j	x_j	f(x_j)
1	0.0	1.000
2	0.1	0.990
3	0.2	0.961
4	0.3	0.914
5	0.4	0.852
6	0.5	0.779
7	0.6	0.698
8	0.7	0.613
9	0.8	0.527
10	0.9	0.445
11	1.0	0.368

The value of the integral \int_{0}^{1} f(x) \, dx using the Simpson's rule is

A

0.5469

B

0.6469

C

0.7469

D

0.8469

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2

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2007

For the following system of equations
\[ \begin{aligned} 4x_1 + x_2 + x_3 &= 4 \\ x_1 + 4x_2 - 2x_3 &= 4 \\ 3x_1 + 2x_2 - 4x_3 &= 6 \end{aligned} \]
which of the following is the solution, obtained after TWO iterations using Jacobi method

A

x_1 = 1.0, x_2 = 1.0, x_3 = 0.0

B

x_1 = 1.0, x_2 = 1.0, x_3 = -1.5

C

x_1 = 1.06, x_2 = 0.594, x_3 = -0.656

D

x_1 = 1.125, x_2 = 0.0, x_3 = -0.25

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3

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2008

The Newton iterative method \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), \(n = 0,1,2,3, ...\), gives the first order approximate root \(x_1\) of the function \(f(x) = x^3 - 6x + 2\) with \(x_0 = 0\) as

A

\(\frac{1}{3}\)

B

\(\frac{1}{6}\)

C

\(\frac{1}{12}\)

D

0

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4

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2008

The trapezoidal rule to evaluate integrals is expressed as \(\int_a^b f(x) dx = \frac{(b-a)}{2} [f(a) + f(b)]\). Using the above expression, evaluate the integral \(\int_0^1 \frac{dx}{1+2x}\) by subdividing the interval \([0,1]\) into two equal parts. The value of this integral is

A

\(\frac{7}{6}\)

B

\(\frac{2}{3}\)

C

\(\frac{7}{12}\)

D

\(\frac{1}{3}\)

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5

2014 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2014

The 51^st common term of the two arithmetic sequences 15, 19, 23 …… and 14, 19, 24 ……, is …….

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6

2014 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2014

\[ \int_0^1 e^x dx \] is evaluated both by Trapezoidal rule and Simpson's 1/3 rule by taking two subintervals. The difference between the results, accurate to the second decimal place, is …….

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7

2016 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2016

Which of the following is a multi-step numerical method for solving the ordinary differential equation?

A

Euler method

B

Improved Euler method

C

Runge-Kutta method

D

Adams-Multon method

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8

2017 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2017

Using Simpson's 1/3 rule, the value of the integral \( \frac{1}{\pi} \int_{0}^{\pi} (1 + \sqrt{\sin(x)}) dx \), accurate to two decimal places, is __________

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9

2018 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2018

Starting from the initial point x₀ = 10, if the sequence {xₙ} is generated using Newton Raphson method to compute the root of the equation x⁴ - 600 = 0, then x₂, accurate to two decimal places, is equal to __________

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10

2019 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2019

The value of the integral \(\frac{6}{\pi} \int_{0}^{\pi/2} \frac{\cos 2x}{1+\sin x} dx\) obtained using Simpson's \(\frac{1}{3}\) rule (rounded off to 2 decimal places) is ____________.

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11

2020 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2020

Assuming the step size \( h = 1 \), the numerical value of the definite integral \( \int_{0}^{2} \frac{x^2}{1+x^2} dx \) obtained using Trapezoidal rule (rounded off to 2 decimal places) is __________.

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12

2021 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2021

If the numerical solution of the initial value problem
\( y' = \frac{t^2}{t + y^3}, \; y(0) = 1, \)
is obtained by the Euler’s method with step size of 0.2, then the value of \( y(0.4) \), (rounded off to two decimal places), is __________.

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13

2022 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2022

Assuming the step size h = 1, the numeric value (rounded off to 2 decimal places) of the definite integral \[ \int_{1}^{3} \frac{x}{1+x} dx \] obtained using Simpson's rule is ______________.

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14

2023 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2023

The Newton-Raphson method is being used for two iterations to find an approximate solution of the equation $e^x - 1 = 0$ with an initial guess of 1. The difference between the actual and approximate solutions (rounded off to 2 decimal places) is __________.

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15

2023 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2023

The area under the curve $y = x^2 + 2x$ between $x = 0$ and $x = 4$, using the trapezoidal rule with a step size of one, (in integer) is __________.

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16

2024 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2024

The approximate value of the integral \[ \int_{2}^{3} \frac{dx}{x} \] using Simpson’s rule with \( h = 0.5 \) is

A

0.12

B

0.19

C

0.41

D

0.58

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17

2024 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2024

A scientist wants to find the root of the equation \(2x^3 + x^2 - 1 = 0\) lying in (0,1). He applies Secant method only once by taking two initial guesses 0.5 and 0.7. The value of the root is approximately

A

0.17

B

0.52

C

0.65

D

0.75

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18

2025 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2025

The value of \(a\) for which the Euler method with step size \(h = 0.1\) provides \(y(1.1) = 1.2\) for the following initial value problem is
\[\frac{dy}{dx} = -2xy^a, \quad y(1) = 2\]

A

-2

B

-1

C

1

D

2

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19

2025 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering & Fibre Science (TF) 2025

For a textile industry, the revenue of selling \( x \) ton of yarn is \( R(x) = 4x \) and the cost of producing \( x \) ton of yarn is \( C(x) = 10 + 2x + 3x^{2/3} \). The industry decided to calculate the break-even (when revenue is equal to the cost) for \( x \) ton of yarn using Newton-Rapson method. Assuming the initial break-even of 8 ton, the break-even (ton) after the first iteration is

A

6

B

8

C

12

D

14

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20

2026 · Textile Engineering & Fibre Science · Engineering Mathematics · Numerical Methods

Textile Engineering and Fibre Science (TF) 2026

The value of the integral \(\int_{0}^{6} x^2 dx\) calculated using the trapezoidal rule with \(n = 2\) (two subintervals) is __________. (answer in integer)

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