Numerical Methods - Engineering Mathematics - Chemical Engineering Previous Year Questions

1

2008 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2008

Q.5 A nonlinear function f(x) is defined in the interval -1.2 < x < 4 as illustrated in the figure below. The equation f(x) = 0 is solved for x within this interval by using the Newton-Raphson iterative scheme. Among the initial guesses (I1, I2, I3, and I4), the guess that is likely to lead to the root most rapidly is

A

\(I_1 \)

B

\(I_2 \)

C

\(I_3 \)

D

\(I_4 \)

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2

2008 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2008

Using Simpson's 1/3 rule and FOUR equally spaced intervals (n = 4), estimate the value of the integral \( \int_{0}^{\pi/4} \frac{\sin x}{\cos^3 x} \, dx \)

A

0.3887

B

0.4384

C

0.5016

D

0.5527

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3

2008 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2008

The following differential equation is to be solved numerically by the Euler's explicit method.
\( \frac{dy}{dx} = x^2 y - 1.2 y \) with \( y(0) = 1 \)
A step size of 0.1 is used. The solution for \( y \) at \( x = 0.1 \) is

A

0.880

B

0.905

C

1.000

D

1.100

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4

2009 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2009

Using the trapezoidal rule and 4 equal intervals (\(n = 4\)), the calculated value of the integral (rounded to the first place of decimal) \(\int_0^{\pi} \sin \theta \, d\theta\) is

A

1.7

B

1.9

C

2.0

D

2.1

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5

2010 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2010

A root of the equation \( x^4 - 3x + 1 = 0 \) needs to be found using the Newton-Raphson method. If the initial guess, \( x_0 \), is taken as 0, then the new estimate, \( x_1 \), after the first iteration is

A

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

B

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

C

3

D

-3

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6

2011 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2011

In the fixed point iteration method for solving equations of the form \( x = g(x) \), the \((n+1)^{th}\) iteration value is \( x_{n+1} = g(x_n) \), where \( x_n \) represents the \( n^{th} \) iteration value. \( g(x) \) and corresponding initial guess value \( x_0 \) in the domain of interest are shown in the following choices. Which ONE of these choices leads to a converged solution for \( x \)?

A

B

C

D

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7

2012 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2012

The Newton-Raphson method is used to find the roots of the equation \[ f(x) = x - \cos \pi x \quad 0 \le x \le 1. \] If the initial guess for the root is 0.5, then the value of \( x \) after the first iteration is

A

1.02

B

0.62

C

0.55

D

0.38

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8

2013 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2013

The value of the integral \(\int_{0.1}^{0.5} e^{-x^3} dx\) evaluated by Simpson's rule using 4 subintervals (up to 3 digits after the decimal point) is ______

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9

2014 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2014

Consider the following differential equation
\(\frac{dy}{dx} = x + \ln(y) ; \quad y = 2 \text{ at } x = 0\)
The solution of this equation at \(x = 0.4\) using Euler method with a step size of \(h = 0.2\) is ____

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10

2017 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2017

Match the problem type in Group-1 with the numerical method in Group-2.
Group-1
P) System of linear algebraic equations
Q) Non-linear algebraic equations
R) Ordinary differential equations
S) Numerical integration
Group-2
I) Newton-Raphson
II) Gauss-Seidel
III) Simpson's rule
IV) Runge-Kutta
Choose the correct set of combinations.

A

P-II, Q-I, R-III, S-IV

B

P-I, Q-II, R-IV, S-III

C

P-IV, Q-III, R-II, S-I

D

P-II, Q-I, R-IV, S-III

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11

2018 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2018

The fourth order Runge-Kutta (RK4) method to solve an ordinary differential equation \(\frac{dy}{dx} = f(x,y)\) is given as
\[y(x+h) = y(x) + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\]
\[k_1 = h f(x, y)\]
\[k_2 = h f\left(x+\frac{h}{2}, y+\frac{k_1}{2}\right)\]
\[k_3 = h f\left(x+\frac{h}{2}, y+\frac{k_2}{2}\right)\]
\[k_4 = h f\left(x+h, y+k_3\right)\]
For a special case when the function \(f\) depends solely on \(x\), the above RK4 method reduces to

A

Euler’s explicit method

B

Trapezoidal rule

C

Euler’s implicit method

D

Simpson’s 1/3 rule

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12

2018 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2018

The volume of liquid filled in a spherical storage tank of radius R is computed from height of liquid, h, in the outside tube (neglecting the volume of liquid in the outside tube) as \( V = \pi h^2 \frac{(3R - h)}{3} \). The estimate of liquid height (in m) to store V = 30 m³ of water in R = 3 m tank, after performing ONE iteration of Secant method, using 1 m and 3 m as two initial guesses of liquid height is __________ (rounded off to second decimal place).

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13

2019 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2019

The Newton-Raphson method is used to determine the root of the equation f(x) = e^–x – x. If the initial guess for the root is 0, the estimate of the root after two iterations is ______ (rounded off to three decimal places).

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14

2020 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2020

Which one of the following methods requires specifying an initial interval containing the root (i.e., bracketing) to obtain the solution of \(f(x) = 0\), where \(f(x)\) is a continuous non-linear algebraic function?

A

Newton-Raphson method

B

regula falsi method

C

secant method

D

fixed point iteration method

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15

2020 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2020

Consider the following dataset.

\(x\)	1	3	5	15	25
\(f(x)\)	6	8	10	12	5

The value of the integral \(\int_{1}^{25} f(x) dx\) using Simpson's 1/3^rd rule is \_\_\_\_\_\_\_\_\_\_ (round off to 1 decimal place).

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16

2021 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2021

An ordinary differential equation (ODE), \(\frac{dy}{dx} = 2y\), with an initial condition \(y(0) = 1\), has the analytical solution \(y = e^{2x}\).
Using Runge-Kutta second order method, numerically integrate the ODE to calculate \(y\) at \(x = 0.5\) using a step size of \(h = 0.5\).
If the relative percentage error is defined as,
\[ \epsilon = \left| \frac{y_{analytical} - y_{numerical}}{y_{analytical}} \right| \times 100 \]
then the value of \(\epsilon\) at \(x = 0.5\) is ________.

A

0.06

B

0.8

C

4.0

D

8.0

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17

2021 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2021

To solve an algebraic equation \( f(x) = 0 \), an iterative scheme of the type
\( x_{n+1} = g(x_n) \) is proposed, where \( g(x) = x - \frac{f(x)}{f'(x)} \).
At the solution \( x = s \), \( g'(s) = 0 \) and \( g''(s) \neq 0 \).
The order of convergence for this iterative scheme near the solution is __________.

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18

2022 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2022

The area of a circular field is 25 m². The radius, r, is to be determined using the Newton-Raphson iterative method. For an initial guess of r = 2.500 m, the revised estimate of r after one iteration is __________ m (rounded off to three decimal places).

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19

2022 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2022

The equation \(\frac{dy}{dx} = xy^2 + 2y + x - 4.5\) with the initial condition \(y(x=0)=1\) is to be solved using a predictor-corrector approach. Use a predictor based on the implicit Euler’s method and a corrector based on the trapezoidal rule of integration, each with a full-step size of 0.5. Considering only positive values of \(y\), the value of \(y\) at \(x=0.5\) is __________ (rounded off to three decimal places).

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20

2023 · Chemical Engineering · Engineering Mathematics · Numerical Methods

Chemical Engineering (CH) 2023

Simpson’s one-third rule is used to estimate the definite integral \[ I = \int_{-1}^{1} \sqrt{1 - x^2} dx \] with an interval length of 0.5. Which one of the following is the CORRECT estimate of \( I \) obtained using this rule?

A

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

B

\(\frac{1}{3} + \frac{2}{\sqrt{3}} \)

C

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

D

\(\frac{1}{3} - \frac{2}{\sqrt{3}} \)

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