Complex Variables - Engineering Mathematics - Petroleum Engineering Previous Year Questions

1

2016 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2016

For a complex number \( Z = \left( \frac{1}{2} + \frac{\sqrt{3}}{2} i \right) \), the value of \( Z^6 \) is

A

- \(\left( \frac{1}{2} + \frac{\sqrt{3}}{2} i \right) \)

B

-1

C

\(\left( \frac{1}{2} - \frac{\sqrt{3}}{2} i \right) \)

D

1

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2

2017 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2017

If \( 5x + 2iy - ix + 7y = 2 + 3i \), where \( i = \sqrt{-1} \), the values of two real numbers (x,y) are, respectively:

A

(-1,1)

B

(1,-1)

C

(1,1)

D

(-1,-1)

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3

2017 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2017

Pick the INCORRECT inequality, where \( z_1 \), \( z_2 \) and \( z_3 \) are complex numbers.

A

\(|z_1 + z_2| \leq |z_1| + |z_2| \)

B

\(|z_1 - z_2| \geq |z_1| - |z_2| \)

C

\(|z_1 - z_2| \leq |z_1| - |z_2| \)

D

\(|z_1 + z_2 + z_3| \leq |z_1| + |z_2| + |z_3| \)

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4

2017 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2017

Which of the following is NOT true? ( \( i = \sqrt{-1} \) )

A

\(\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \)

B

\(e^{i\theta} = \cos \theta + i \sin \theta \)

C

\(\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \)

D

\(\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2i} \)

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5

2017 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2017

\(z = \frac{3i^{30} - i^{19}}{2i - 1}\), where \(i = \sqrt{-1}\), would simplify to:

A

\(1 - i\)

B

1

C

\(-i\)

D

\(1 + i\)

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6

2018 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2018

Two complex numbers, \(\tilde{k}\) and \(\tilde{\varepsilon}\) are related as follows: \[ \tilde{k} = \frac{\tilde{\varepsilon}}{i\omega} \] where, \(i = \sqrt{-1}\) and \(\omega\) is a scalar. Given principal argument of \(\tilde{\varepsilon}\), \(\text{Arg}(\tilde{\varepsilon}) = -\frac{2\pi}{3}\), the principal argument of \(\tilde{k}\), \(\text{Arg}(\tilde{k}) = \)__________. (rounded-off to two decimal places. Use \(\pi = 3.14\))

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7

2019 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2019

For a complex number \( z = 1 - 4i \) with \( i = \sqrt{-1} \), the value of \( \left| \frac{z+3}{z-1} \right| \) is

A

0

B

1/\sqrt{2}

C

1

D

\(\sqrt{2}\)

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8

2019 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2019

If \(z\) is a complex variable with \(i = \sqrt{-1}\), the length of the minor axis of an ellipse defined by \(|z - (1+i)| + |z - (9+i)| = 10\) is ______

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9

2020 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2020

Two complex numbers are given as, \( z_1 = e^{i\theta_1} \) and \( z_2 = e^{i\theta_2} \), where, \( i = \sqrt{-1} \), and \( \theta_1 \) and \( \theta_2 \) are the principal arguments. Given, \( \theta_1 \neq \theta_2 \) and \( |\theta_1 - \theta_2| \neq \pi \). If \( m = \sqrt{(\cos \theta_1 + \cos \theta_2)^2 + (\sin \theta_1 + \sin \theta_2)^2} \), which one of the following conditions is correct?

A

\(2 < m < 3 \)

B

\(0 < m < 2 \)

C

\(m = 2 \)

D

\(m = 0 \)

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10

2021 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2021

Given two complex numbers, \( z_1 = 4 + 3i \) and \( z_2 = 2 - 5i \), the real part of \( z_1 z_2 \) is __________.

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11

2024 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2024

A complex number is defined as \( z = x + iy \) with \( i = \sqrt{-1} \). \( \bar{z} \) is the complex conjugate of \( z \). The imaginary part of \( (2z + 4\bar{z} + 4iy) \) is ________.

A

6

B

2

C

2y

D

3y

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12

2023 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2023

Let \( z_1 \) and \( z_2 \) be two arbitrary complex numbers with non-zero modulus. Which of the following conditions is FALSE?

A

\(|z_1 + z_2| > |z_1| + |z_2| \)

B

\(0 \leq |z_1 + z_2| < \infty \)

C

\(|z_1 + z_2| \leq |z_1| + |z_2| \)

D

\(|z_1 z_2| = |z_1||z_2| \)

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13

2026 · Petroleum Engineering · Engineering Mathematics · Complex Variables

Petroleum Engineering (PE) 2026

Which of the following is the correct representation of the given expression? \[ \frac{(1 + i)^8}{(1 - i)^6} \]

A

2e^{-i\frac{\pi}{2}}

B

-1

C

2e^{i\frac{\pi}{2}}

D

1

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