Complex Variable MCQs Question and Answer in Engineering Maths

51

The value of the integral $$\oint {\frac{{{\text{z}} + 1}}{{{{\text{z}}^2} - 4}}{\text{dz}}} $$ in counter clockwise direction around a circle C of radius 1 with center at the point z = -2 is

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

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Answer & Solution

Answer: Option A

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52

The integral $$\oint {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ evaluated around the unit circle on the complex plane for $${\text{f}}\left( {\text{z}} \right) = \frac{{\cos {\text{z}}}}{{\text{z}}}$$ is

A) 2πi B) 4πi C) -2πi D) 0

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Answer: Option A

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53

The residues of a function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 4} \right){{\left( {{\text{z}} + 1} \right)}^3}}}$$ are

A) $$\frac{{ - 1}}{{27}}{\text{ and }}\frac{{ - 1}}{{125}}$$ B) $$\frac{1}{{125}}{\text{ and }}\frac{{ - 1}}{{125}}$$ C) $$\frac{{ - 1}}{{27}}{\text{ and }}\frac{1}{5}$$ D) $$\frac{1}{{125}}{\text{ and }}\frac{{ - 1}}{5}$$

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Answer: Option B

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54

The value of expression $$\frac{{ - 5 + {\text{i}}10}}{{3 + 4{\text{i}}}}$$ is

A) 1 - 2i B) 1 + 2i C) 2 - i D) 2 + i

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Answer: Option B

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55

The argument of the complex number $$\frac{{1 + {\text{i}}}}{{1 - {\text{i}}}},$$ where $${\text{i}} = \sqrt { - 1} ,$$ is

A) $$ - \pi $$ B) $$ - \frac{\pi }{2}$$ C) $$\frac{\pi }{2}$$ D) $$\pi $$

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Answer & Solution

Answer: Option C

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56

The nature of singularity of function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\cos {\text{z}} - \sin {\text{z}}}}$$ at $${\text{z}} = \frac{\pi }{4}$$ is

A) Removable singularity B) Isolated singularity C) Simple pole D) Essential singularity

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Answer: Option C

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57

A point z has been plotted in the complex plane, as shown in figure below.
Complex Variable mcq question image

The plot for point $$\frac{1}{{\text{z}}}$$ is

A)

B)

C)

D)

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Answer & Solution

Answer: Option D

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58

The analytic function $${\text{f}}\left( {\text{z}} \right) = \frac{{{\text{z}} - 1}}{{{{\text{z}}^2} + 1}}$$ has singularities at

A) 1 and -1 B) 1 and i C) 1 and -i D) i and -i

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Answer: Option D

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59

The values of the integral $$\frac{1}{{2\pi {\text{j}}}}\oint\limits_{\text{c}} {\frac{{{{\text{e}}^{\text{z}}}}}{{{\text{z}} - 2}}{\text{dz}}} $$ along a closed contour c in anti-clockwise direction for
i. the point z₀ = 2 inside the contour c, and
ii. the point z₀ = 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

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Answer: Option B

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60

Evaluate: $$\int\limits_{\text{c}} {\frac{{{\text{dz}}}}{{{\text{z}}\sin {\text{z}}}}} ,$$ where c is x² + y² = 1.

A) 1 B) 2 C) 0 D) -1

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Answer: Option C

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