Complex Variable MCQs Question and Answer in Engineering Maths Page-3 section-1

21.
If C is a circle |z| = 4 and $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2}}}{{{{\left( {{{\text{z}}^2} - 3{\text{z}} + 2} \right)}^2}}},$$ then $$\oint\limits_{\text{c}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is

A. 1

B. 0

C. -1

D. -2

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22.
What is the residue of the function $$\frac{{1 - {{\text{e}}^{2{\text{z}}}}}}{{{{\text{z}}^4}}}$$ at its pole?

A. $$\frac{4}{3}$$

B. $$ - \frac{4}{3}$$

C. $$ - \frac{2}{3}$$

D. $$\frac{2}{3}$$

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23.
Let z = x + jy where $${\text{j}} = \sqrt { - 1} .$$ Then $$\overline {\cos {\text{z}}} = ?$$

A. cos z

B. cos $$\overline {\text{z}} $$

C. sin z

D. sin $$\overline {\text{z}} $$

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24.
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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25.
Consider the analytic function f(z) = x² - y² + i2xy of the complex variable z = x + iy, where $${\text{i}} = \sqrt { - 1} .$$ The derivative f'(z) is

A. 2x + i2y

B. x² + iy²

C. x + iy

D. 2x - i2y

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26.
Using Cauchy's integral theorem, the value of the integral (integration being taken in counterclockwise direction) $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^3} - 6}}{{3{\text{z}} - {\text{i}}}}{\text{dz}}} $$ is

A. $$\frac{{2\pi }}{{81}} - 4\pi {\text{i}}$$

B. $$\frac{\pi }{8} - 6\pi {\text{i}}$$

C. $$\frac{{4\pi }}{{81}} - 6\pi {\text{i}}$$

D. 1

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27.
The value of the integral $$\int\limits_{\text{c}} {\frac{{\cos \left( {2\pi {\text{z}}} \right)}}{{\left( {2{\text{z}} - 1} \right)\left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ (where C is a closed curve given by |z| = 1) is

A. $$ - \pi {\text{i}}$$

B. $$\frac{{\pi {\text{i}}}}{5}$$

C. $$\frac{{2\pi {\text{i}}}}{5}$$

D. $$\pi {\text{i}}$$

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28.
If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $$\oint\limits_{\text{D}} {\frac{1}{{\left( {{{\text{s}}^2} - 1} \right)}}{\text{ds}}} $$ is

A. jπ

B. -jπ

C. -π

D. π

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29.
A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x² - 2y² + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

A. 4y² - 4xy + constant

B. -4xy + 2y² - 2x² + constant

C. 2x² - 2y² + xy + constant

D. 4xy - 2x² + 2y² + constant

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30.
The complex function tan h(s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n

A. Re(s) = 0

B. $$I$$m(s) ≠ nπ

C. $$I{\text{m}}\left( {\text{s}} \right) \ne \frac{{{\text{n}}\pi }}{3}$$

D. $$I{\text{m}}\left( {\text{s}} \right) \ne \frac{{\left( {2{\text{n}} + 1} \right)\pi }}{2}$$

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Complex Variable MCQs Question and Answer in Engineering Maths

21. If C is a circle |z| = 4 and $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2}}}{{{{\left( {{{\text{z}}^2} - 3{\text{z}} + 2} \right)}^2}}},$$ then $$\oint\limits_{\text{c}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is

Answer & Solution

22. What is the residue of the function $$\frac{{1 - {{\text{e}}^{2{\text{z}}}}}}{{{{\text{z}}^4}}}$$ at its pole?

Answer & Solution

23. Let z = x + jy where $${\text{j}} = \sqrt { - 1} .$$ Then $$\overline {\cos {\text{z}}} = ?$$

Answer & Solution

24. 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?

Answer & Solution

25. Consider the analytic function f(z) = x2 - y2 + i2xy of the complex variable z = x + iy, where $${\text{i}} = \sqrt { - 1} .$$ The derivative f'(z) is

Answer & Solution

26. Using Cauchy's integral theorem, the value of the integral (integration being taken in counterclockwise direction) $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^3} - 6}}{{3{\text{z}} - {\text{i}}}}{\text{dz}}} $$ is

Answer & Solution

27. The value of the integral $$\int\limits_{\text{c}} {\frac{{\cos \left( {2\pi {\text{z}}} \right)}}{{\left( {2{\text{z}} - 1} \right)\left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ (where C is a closed curve given by |z| = 1) is

Answer & Solution

28. If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $$\oint\limits_{\text{D}} {\frac{1}{{\left( {{{\text{s}}^2} - 1} \right)}}{\text{ds}}} $$ is

Answer & Solution

29. A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 - 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

Answer & Solution

30. The complex function tan h(s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n

Answer & Solution

21.
If C is a circle |z| = 4 and $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2}}}{{{{\left( {{{\text{z}}^2} - 3{\text{z}} + 2} \right)}^2}}},$$ then $$\oint\limits_{\text{c}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is

22.
What is the residue of the function $$\frac{{1 - {{\text{e}}^{2{\text{z}}}}}}{{{{\text{z}}^4}}}$$ at its pole?

23.
Let z = x + jy where $${\text{j}} = \sqrt { - 1} .$$ Then $$\overline {\cos {\text{z}}} = ?$$

24.
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?

25.
Consider the analytic function f(z) = x² - y² + i2xy of the complex variable z = x + iy, where $${\text{i}} = \sqrt { - 1} .$$ The derivative f'(z) is

26.
Using Cauchy's integral theorem, the value of the integral (integration being taken in counterclockwise direction) $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^3} - 6}}{{3{\text{z}} - {\text{i}}}}{\text{dz}}} $$ is

27.
The value of the integral $$\int\limits_{\text{c}} {\frac{{\cos \left( {2\pi {\text{z}}} \right)}}{{\left( {2{\text{z}} - 1} \right)\left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ (where C is a closed curve given by |z| = 1) is

28.
If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $$\oint\limits_{\text{D}} {\frac{1}{{\left( {{{\text{s}}^2} - 1} \right)}}{\text{ds}}} $$ is

29.
A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x² - 2y² + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

30.
The complex function tan h(s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n