Complex Variable MCQs Question and Answer in Engineering Maths

1

f(z) = u(x, y) + iv(x, y) is an analytic function or complex variable z = x + iy where $${\text{i}} = \sqrt { - 1} ,$$ u(x, y) = 2xy, then v(x, y) may be expressed as

A) -x² + y² + constant B) x² - y² + constant C) x² + y² + constant D) -(x² + y²) + constant

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Answer & Solution

Answer: Option A

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2

The product of complex numbers (3 - 2i) and (3 + i4) results in

A) 1 + 6i B) 9 - 8i C) 9 + 8i D) 17 + 6i

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Answer: Option D

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3

If a complex number $${\text{z}} = \frac{{\sqrt 3 }}{2} + {\text{i}}\frac{1}{2}$$ then z⁴ is

A) $$2\sqrt 2 + 2{\text{i}}$$ B) $$\frac{{ - 1}}{2} + \frac{{{\text{i}}{{\sqrt 3 }^2}}}{2}$$ C) $$\frac{{\sqrt 3 }}{2} - {\text{i}}\frac{1}{2}$$ D) $$\frac{{\sqrt 3 }}{2} - {\text{i}}\frac{1}{8}$$

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Answer & Solution

Answer: Option B

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4

If C is a circle of radius r with centre z₀, in the complex z-plane and if n is a non-zero integer, then $$\oint {\frac{{{\text{dz}}}}{{{{\left( {{\text{z}} - {{\text{z}}_0}} \right)}^{{\text{n}} + 1}}}}} $$ equals

A) 2πnj B) 0 C) $$\frac{{\pi {\text{j}}}}{{2\pi }}$$ D) 2πn

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Answer & Solution

Answer: Option B

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5

Contour C in the adjoining figure is described by x² + y² = 16. The value $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^2} + 8}}{{0.5{\text{z}} - 1.5{\text{j}}}}{\text{dz}}} $$ is (Note: $${\text{j}} = \sqrt { - 1} $$ )
Complex Variable mcq question image

A) -2πj B) 2πj C) 4πj D) -4πj

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Answer & Solution

Answer: Option D

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6

The value of $$\oint {\frac{{\sin {\text{z}}}}{{\text{z}}}{\text{dz,}}} $$ where the counter of integration is a simple closed curve around the origin, is

A) 0 B) 2πj C) $$\infty $$ D) $$\frac{1}{{2\pi {\text{j}}}}$$

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Answer & Solution

Answer: Option A

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7

$${\text{z}} = \frac{{2 - 3{\text{j}}}}{{ - 5 + {\text{i}}}}$$ can be expressed as

A) -0.5 - 0.5i B) -0.5 + 0.5i C) 0.5 - 0.5i D) 0.5 + 0.5i

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Answer & Solution

Answer: Option B

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8

Consider the following complex function $${\text{f}}\left( {\text{z}} \right) = \frac{9}{{\left( {{\text{z}} - 1} \right){{\left( {{\text{z}} + 2} \right)}^2}}}$$
Which of the following is one of the residues of the above function?

A) -1 B) $$\frac{9}{{16}}$$ C) 2 D) 9

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Answer & Solution

Answer: Option A

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9

The value of $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^2}}}{{{{\text{z}}^4} - 1}}{\text{dz}}} $$ using Cauchy's integral formula, around circle |z + 1| = 1 where z = x + iy is

A) 2πi B) $$ - \pi \frac{{\text{i}}}{2}$$ C) $$ - 3\pi \frac{{\text{i}}}{2}$$ D) π²i

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Answer: Option B

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10

The modulus of the complex number $$\left( {\frac{{3 + 4{\text{i}}}}{{1 - 2{\text{i}}}}} \right)$$ is

A) 5 B) √5 C) $$\frac{1}{{\sqrt 5 }}$$ D) $$\frac{1}{5}$$

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Answer & Solution

Answer: Option B

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