Complex Variables - Engineering Mathematics - Electrical Engineering Previous Year Questions

1

2007 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2007

The value of ∫_C dz/(1+z²) where C is the contour |z - i/2| = 1 is

A

2πi

B

π

C

tan⁻¹ z

D

πi tan⁻¹ z

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2

2011 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2011

Roots of the algebraic equation \(x^3+x^2+x+1=0\) are

A

\(+1, +j, -j\)

B

\(+1, -1, +1\)

C

\((0,0,0)\)

D

\(-1, +j, -j\)

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3

2011 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2011

A point z has been plotted in the complex plane, as shown in figure below.

The plot of the complex number \( y = \frac{1}{z} \) is

A

Plot (A)

B

Plot (B)

C

Plot (C)

D

Plot (D)

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4

2012 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2012

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

A

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

B

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

C

x

D

1

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5

2012 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 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 \(\frac{1}{2\pi j} \oint_C f(z)dz\) is

A

-2

B

-1

C

1

D

2

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6

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 1]

Square roots of \(-i\), where \( i = \sqrt{-1} \), are

A

\(i, -i \)

B

\(\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4}), \cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}) \)

C

\(\cos(\frac{\pi}{4}) + i \sin(\frac{3\pi}{4}), \cos(\frac{3\pi}{4}) + i \sin(\frac{\pi}{4}) \)

D

\(\cos(\frac{3\pi}{4}) + i \sin(-\frac{3\pi}{4}), \cos(-\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}) \)

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7

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 1]

$\oint \frac{z^2 - 4}{z^2 + 4} dz$ evaluated anticlockwise around the circle $|z - i| = 2$, where $i = \sqrt{-1}$, is

A

\(-4\pi\)

B

0

C

2 + \pi

D

2 + 2i

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8

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 2]

Square roots of \( -i \), where \( i = \sqrt{-1} \), are

A

\(i, -i \)

B

\(\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4}), \cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}) \)

C

\(\cos(\frac{\pi}{4}) + i \sin(\frac{3\pi}{4}), \cos(\frac{3\pi}{4}) + i \sin(\frac{\pi}{4}) \)

D

\(\cos(\frac{3\pi}{4}) + i \sin(-\frac{3\pi}{4}), \cos(-\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}) \)

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9

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 2]

∮ (z² - 4) / (z² + 4) dz evaluated anticlockwise around the circle |z - i| = 2, where i = √-1, is

A

-4π

B

0

C

2 + π

D

2 + 2i

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10

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 3]

Square roots of \(-i\), where \(i=\sqrt{-1}\), are

A

\(i, -i\)

B

\(\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4}), \cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4})\)

C

\(\cos(\frac{\pi}{4}) + i \sin(\frac{3\pi}{4}), \cos(\frac{3\pi}{4}) + i \sin(\frac{\pi}{4})\)

D

\(\cos(\frac{3\pi}{4}) + i \sin(-\frac{3\pi}{4}), \cos(-\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4})\)

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11

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 3]

\(\oint \frac{z^2-4}{z^2+4}dz\) evaluated anticlockwise around the circle \(|z-i|=2\), where \(i=\sqrt{-1}\), is

A

\(-4\pi\)

B

0

C

\(2+\pi\)

D

\(2+2i\)

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12

2013 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2013 [Session 4]

\(\oint_{|z-i|=2} \frac{z^2-4}{z^2+4} dz\) evaluated anticlockwise around the circle \(|z-i|=2\), where \(i=\sqrt{-1}\), is

A

\(-4\pi\)

B

0

C

2+\pi

D

2+2i

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13

2014 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2014 [Session 1]

Let \( S \) be the set of points in the complex plane corresponding to the unit circle. (That is, \( S = \{ z : |z| = 1 \} \). Consider the function \( f(z) = z z^* \) where \( z^* \) denotes the complex conjugate of \( z \). The \( f(z) \) maps \( S \) to which one of the following in the complex plane?

A

unit circle

B

horizontal axis line segment from origin to (1, 0)

C

the point (1, 0)

D

the entire horizontal axis

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14

2014 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2014 [Session 2]

All the values of the multi-valued complex function \( 1^i \), where \( i = \sqrt{-1} \), are

A

purely imaginary.

B

real and non-negative.

C

on the unit circle.

D

equal in real and imaginary parts.

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15

2014 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2014 [Session 3]

Integration of the complex function \( f(z) = \frac{z^2}{z^2-1} \), in the counterclockwise direction, around \( |z-1| = 1 \), is

A

\(-\pi i \)

B

0

C

\(\pi i \)

D

\(2\pi i \)

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16

2016 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2016 [Session 1]

The value of the integral \(\oint_C \frac{2z + 5}{(z - \frac{1}{2})(z^2 - 4z + 5)} dz\) over the contour \(|z| = 1\), taken in the anti-clockwise direction, would be

A

\(\frac{24\pi i}{13}\)

B

\(\frac{48\pi i}{13}\)

C

\(\frac{24}{13}\)

D

\(\frac{12}{13}\)

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17

2016 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2016 [Session 2]

Consider the function \(f(z) = z + z^*\) where \(z\) is a complex variable and \(z^*\) denotes its complex conjugate. Which one of the following is TRUE?

A

\(f(z)\) is both continuous and analytic

B

\(f(z)\) is continuous but not analytic

C

\(f(z)\) is not continuous but is analytic

D

\(f(z)\) is neither continuous nor analytic

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18

2017 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2017 [Session 1]

For a complex number \(z\), \(\lim_{z \to i} \frac{z^2+1}{z^3+2z-i(z^2+2)}\) is

A

\(-2i\)

B

\(-i\)

C

\(i\)

D

\(2i\)

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19

2017 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2017 [Session 1]

Consider the line integral $I = \int_c (x^2 + iy^2)dz$, where $z = x + iy$. The line $c$ is shown in the figure below.

The value of $I$ is

A

$\frac{1}{2}i$

B

$\frac{2}{3}i$

C

$\frac{3}{4}i$

D

$\frac{4}{5}i$

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20

2017 · Electrical Engineering · Engineering Mathematics · Complex Variables

Electrical Engineering (EE) 2017 [Session 2]

The value of the contour integral in the complex-plane \[ \oint \frac{z^3 - 2z + 3}{z-2} \, dz \] along the contour $|z| = 3$, taken counter-clockwise is

A

$-18\pi i$

B

0

C

$14\pi i$

D

$48\pi i$

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