Complex Variables - Engineering Mathematics - Chemical Engineering Previous Year Questions

1

2007 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2007

Given \(i = \sqrt{-1}\), the ratio \(\frac{(i+3)}{(i+1)}\) is given by

A

\(i\)

B

\(-2\)

C

\(-i+2\)

D

\(i+1\)

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2

2007 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2007

If \( z = x + iy \) is a complex number, where \( i = \sqrt{-1} \) then the derivative of \( z\bar{z} \) at \( 2 + i \) is

A

0

B

2

C

4

D

does not exist

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3

2008 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2008

An analytic function \( w(z) \) is defined as \( w = u + iv \), where \( i = \sqrt{-1} \) and \( z = x + iy \). If the real part is given by \( u = \frac{-y}{x^2 + y^2} \), \( w(z) \) is

A

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

B

\(\frac{1}{z^2} \)

C

\(\frac{i}{z} \)

D

\(\frac{1}{iz} \)

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4

2009 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2009

The modulus of the complex number \( \frac{1+i}{\sqrt{2}} \) is

A

\(\frac{1}{2} \)

B

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

C

1

D

\(\sqrt{2} \)

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5

2009 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2009

Using the residue theorem, the value of the integral (counterclockwise) \(\oint \frac{8 - 7z}{z^2 - 4} dz\) around a circle with center at \(z = 0\) and radius = 8 (where \(z\) is a complex number and \(i = \sqrt{-1}\)), is

A

\(-20\pi i\)

B

\(-40\pi\)

C

\(-40\pi i\)

D

\(40\pi i\)

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6

2010 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2010

Given that \( i = \sqrt{-1} \), \( i^i \) is equal to

A

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

B

\(-1 \)

C

\(i \ln i \)

D

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

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7

2013 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2013

For the function \(f(z) = \frac{1}{(2-z)(z+2)}\), the residue at z = 2 is __________

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8

2014 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2014

If \( f^*(x) \) is the complex conjugate of \( f(x) = \cos(x) + i \sin(x) \), then for real \( a \) and \( b \), \( \int_{a}^{b} f^*(x) f(x) dx \) is ALWAYS

A

positive

B

negative

C

real

D

imaginary

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9

2016 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2016

What are the modulus (\(r\)) and argument (\( heta\)) of the complex number \(3+4i\)?

A

\(r = \sqrt{7},\ \theta = \tan^{-1}(\frac{4}{3})\)

B

\(r = \sqrt{7},\ \theta = \tan^{-1}(\frac{3}{4})\)

C

\(r = 5,\ \theta = \tan^{-1}(\frac{3}{4})\)

D

\(r = 5,\ \theta = \tan^{-1}(\frac{4}{3})\)

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10

2017 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2017

The real part of \( 6e^{i\pi/3} \) is __________.

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11

2019 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2019

The value of the complex number \(i^{-1/2}\) (where \(i = \sqrt{-1}\)) is

A

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

B

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

C

\(\frac{1}{\sqrt{2}}i\)

D

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

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12

2022 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2022

The value of \((1 + i)^{12}\), where \(i = \sqrt{-1}\), is

A

\(-64i\)

B

\(64i\)

C

\(64\)

D

\(-64\)

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13

2023 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2023

Given that \[ F = \frac{|z_1 + z_2|}{|z_1| + |z_2|} , \] where \( z_1 = 2 + 3i \) and \( z_2 = -2 + 3i \) with \( i = \sqrt{-1} \), which one of the following options is CORRECT?

A

\(F < 0 \)

B

\(F < 1 \)

C

\(F > 1 \)

D

\(F = 1 \)

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14

2024 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2024

If \(z_1 = -1 + i\) and \(z_2 = 2i\), where \(i = \sqrt{-1}\), then \(\text{Arg}(z_1/z_2)\) is

A

\(\frac{3\pi}{4}\)

B

\(\frac{\pi}{4}\)

C

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

D

\(\frac{\pi}{3}\)

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15

2025 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2025

Consider two complex numbers \(z_1 = 1 - i\) and \(z_2 = i\). The argument of \(z_1 z_2\) is

A

0

B

\(\frac{\pi}{4}\)

C

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

D

\(\pi\)

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16

2026 · Chemical Engineering · Engineering Mathematics · Complex Variables

Chemical Engineering (CH) 2026

Consider the following complex numbers
\[ z_1 = r_1(\cos \theta_1 + i \sin \theta_1) \]
\[ z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \]
where \(r_1, r_2\) are real numbers, \(0 \leq \theta_1 \leq \frac{\pi}{2}\), \(0 \leq \theta_2 \leq \frac{\pi}{2}\), and \(i = \sqrt{-1}\).
If \(|z_1 + z_2| = |z_1| + |z_2|\), which one of the following conditions is necessarily CORRECT?

A

\(\theta_1 = 0, \theta_2 = \frac{\pi}{2}\)

B

\(\theta_1 = \frac{\pi}{2}, \theta_2 = 0\)

C

\(r_1 = r_2\)

D

\(\theta_1 = \theta_2\)

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