Complex Analysis - Engineering Mathematics - Electronics & Communication Engineering Previous Year Questions

1

2012 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2012

Given \( f(z) = \frac{1}{z+1} - \frac{2}{z+3} \). If \( C \) is a counterclockwise path in the \( z \)-plane such that \( |z+1| = 1 \), the value of \( \frac{1}{2\pi j} \int_C f(z) dz \) is

A

-2

B

-1

C

1

D

2

Open complete paper

2

2012 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2012

If \( x = \sqrt{-1} \), then the value of \( x^x \) is

A

\(e^{-\pi/2} \)

B

\(e^{\pi/2} \)

C

x

D

1

Open complete paper

3

2014 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2014 [Session 1]

\(C\) is a closed path in the z-plane given by \(|z| = 3\). The value of the integral \(\oint_C \left(\frac{z^2 - z + 4j}{z + 2j}\right) dz\) is

A

\(-4\pi(1 + j2)\)

B

\(4\pi(3 - j2)\)

C

\(-4\pi(3 + j2)\)

D

\(4\pi(1 - j2)\)

Open complete paper

4

2014 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2014 [Session 2]

The real part of an analytic function $f(z)$ where $z = x + jy$ is given by $e^{-y} \cos(x)$. The imaginary part of $f(z)$ is

A

$e^{y} \cos(x)$

B

$e^{-y} \sin(x)$

C

$-e^{y} \sin(x)$

D

$-e^{-y} \sin(x)$

Check answer

Open complete paper

5

2015 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 3]

If C is a circle of radius r with centre \( z_0 \), in the complex z-plane and if n is a non-zero integer, then \( \oint_C \frac{dz}{(z-z_0)^{n+1}} \) equals

A

\(2\pi n j \)

B

0

C

\(\frac{n j}{2\pi} \)

D

\(2\pi n \)

Open complete paper

6

2016 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2016 [Session 1]

In the following integral, the contour C encloses the points 2 and −2. \[-\frac{1}{2\pi j}\oint_C \frac{\sin z}{(z - 2\pi j)^3} dz\] The value of the integral is ______

Check answer

Open complete paper

7

2016 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2016 [Session 3]

The values of the integral \(\frac{1}{2 \pi j} \oint_c \frac{e^z}{z-2} d z\) along a closed contour \(c\) in anti-clockwise direction for
(i) the point \(z_0 = 2\) inside the contour \(c\), and
(ii) the point \(z_0 = 2\) outside the contour \(c\),
respectively, are

A

(i) 2.72, (ii) 0

B

(i) 7.39, (ii) 0

C

(i) 0, (ii) 2.72

D

(i) 0, (ii) 7.39

Open complete paper

8

2016 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2016 [Session 3]

For \( f(z) = \frac{\sin(z)}{z^2} \), the residue of the pole at z = 0 is __________

Check answer

Open complete paper

9

2017 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2017

The residues of a function f(z) = 1/((z-4)(z+1)3) are

A

-1/27 and -1/125

B

1/125 and 1/125

C

-1/27 and 1/5

D

1/125 and -1/5

Open complete paper

10

2017 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2017

An integral I over a counterclockwise circle C is given by I = ∮C (z2 - 1)/(z2 + 1) ez dz. If C is defined as |z| = 3, then the value of I is

A

−πi sin(1)

B

−2πi sin(1)

C

−3πi sin(1)

D

−4πi sin(1)

Open complete paper

11

2018 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2018

The contour C given below is on the complex plane z = x + jy, where j = √-1.
The value of the integral \[ \frac{1}{\pi j} \oint_C \frac{dz}{z^2 - 1} \] is ______.

Check answer

Open complete paper

12

2019 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2019

Which one of the following functions is analytic over the entire complex plane?

A

\(\ln(z)\)

B

\(e^{1/z}\)

C

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

D

\(\cos(z)\)

Open complete paper

13

2019 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2019

The value of the contour integral
\[ \frac{1}{2\pi j} \oint \left( z + \frac{1}{z} \right)^2 dz \]
evaluated over the unit circle \( |z| = 1 \) is ______.

Check answer

Open complete paper

14

2021 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2021

Consider the integral \[ \oint_{C} \frac{\sin(x)}{x^2(x^2+4)} \, dx \] where \( C \) is a counter-clockwise oriented circle defined as \( |x - i| = 2 \). The value of the integral is

A

\(-\frac{\pi}{8} \sin(2i)\)

B

\(\frac{\pi}{8} \sin(2i)\)

C

\(-\frac{\pi}{4} \sin(2i)\)

D

\(\frac{\pi}{4} \sin(2i)\)

Open complete paper

15

2022 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2022

A simple closed path \(C\) in the complex plane is shown in the figure. If \[\oint_C \frac{2z}{z^2 - 1} dz = -i\pi A,\] where \(i = \sqrt{-1}\), then the value of \(A\) is ________ (rounded off to two decimal places).

Check answer

Open complete paper

16

2023 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2023

Let \( w^4 = 16j \). Which of the following cannot be a value of \( w \)?

A

\(2e^{\frac{j2\pi}{8}} \)

B

\(2e^{\frac{j\pi}{8}} \)

C

\(2e^{\frac{j5\pi}{8}} \)

D

\(2e^{\frac{j9\pi}{8}} \)

Open complete paper

17

2023 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2023

The value of the contour integral, \( \oint_C \left( \frac{z+2}{z^2+2z+2} \right) dz \), where the contour C is \( \left\{ z: \left| z + 1 - \frac{1}{2}j \right| = 1 \right\} \), taken in the counter clockwise direction, is

A

\(-\pi(1+j) \)

B

\(\pi(1+j) \)

C

\(\pi(1-j) \)

D

\(-\pi(1-j) \)

Open complete paper

18

2024 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2024

Let \( z \) be a complex variable. If \( f(z) = \frac{\sin(\pi z)}{z^2(z-2)} \) and \( C \) is the circle in the complex plane with \( |z|=3 \) then \( \oint_C f(z)dz \) is ________.

A

\(\pi^2 j \)

B

\(j\pi\left(\frac{1}{2}-\pi\right) \)

C

\(j\pi\left(\frac{1}{2}+\pi\right) \)

D

\(-\pi^2 j \)

Open complete paper

19

2025 · Electronics & Communication Engineering · Engineering Mathematics · Complex Analysis

Electronics & Communication Engineering (EC) 2025

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle \(C\) in the complex plane is/are TRUE?

A

\(\oint_C e^z \, dz = 0\)

B

\(\oint_C z^n \, dz = 0\), where \(n\) is an even integer

C

\(\oint_C \cos z \, dz \neq 0\)

D

\(\oint_C \sec z \, dz \neq 0\)

Check answer

Open complete paper