Complex Variable MCQs Question and Answer in Engineering Maths Page-4 section-1

31.
Let f₁(z) = z² and f₂(z) = $$\overline {\text{z}} $$ be two complex variable function. Here $$\overline {\text{z}} $$ is the complex conjugate of z. Choose the correct answer.

A. Both f₁(z) and f₂(z) are analytic

B. Only f₁(z) is analytic

C. Only f₂(z) is analytic

D. Both f₁(z) and f₂(z) are not analytic

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32.
If $${\text{W}} = \phi + {\text{i}}\psi $$ represents the complex potential for an electric field. Given $$\psi = {{\text{x}}^2} - {{\text{y}}^2} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}},$$ then the function $$\phi $$ is

A. $$ - 2{\text{xy}} + \frac{{\text{y}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$

B. $$2{\text{xy}} + \frac{{\text{y}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$

C. $$ - 2{\text{xy}} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$

D. $$2{\text{xy}} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$

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33.
An integral $$I$$ over a counter-clockwise circle C is given by $$I = \oint_{\text{C}} {\frac{{{{\text{z}}^2} - 1}}{{{{\text{z}}^2} + 1}}{{\text{e}}^{\text{z}}}{\text{dz}}{\text{.}}} $$
If C is defined as |z| = 3, then the value of $$I$$ is

A. -πi sin(1)

B. -2πi sin(1)

C. -3πi sin(1)

D. -4πi sin(1)

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34.
The function $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^2} + 4}}$$ is singular at

A. z = ±2

B. z = ±1

C. z = ±i

D. z = ±2i

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35.
If f(z) has a pole of order n at z = a, then Residue of function f(z) at a is

A. $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {\text{n}} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{{\text{n}} - 1}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$

B. $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {{\text{n}} - 1} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{{\text{n}} - 1}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$

C. $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {\text{n}} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{\text{n}}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$

D. $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {{\text{n}} - 1} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{\text{n}}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$

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36.
Which one of the following functions is analytic over the entire complex plane?

A. cos(z)

B. $${{\text{e}}^{\frac{1}{{\text{z}}}}}$$

C. $$l$$n(z)

D. $$\frac{1}{{1 - {\text{z}}}}$$

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37.
Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

A. The residue of $$\frac{{\text{z}}}{{{{\text{z}}^2} - 1}}$$ at z = 1 is $$\frac{1}{2}$$

B. $$\oint_{\text{c}} {{{\text{z}}^2}{\text{dz}} = 0} $$

C. $$\frac{1}{{2\pi {\text{i}}}}\oint_{\text{c}} {\frac{1}{{\text{z}}}\,{\text{dz}} = 1} $$

D. $$\overline {\text{z}} $$ (complex conjugate of z) is analytical function

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38.
An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y), where $${\text{i}} = \sqrt { - 1} .$$ If u(x, y) = 2xy, then v(x, y) must be

A. x² + y² + constant

B. x² - y² + constant

C. -x² + y² + constant

D. -x² - y² + constant

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39.
Polar form of the Cauchy-Reimann equations is

A. $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = {\text{r}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - {\text{r}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$

B. $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = \frac{1}{{\text{r}}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - \frac{1}{{\text{r}}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$

C. $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = \frac{1}{{\text{r}}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - {\text{r}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$

D. $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = {\text{r}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - \frac{1}{{\text{r}}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$

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40.
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) = zz^∗ 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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Complex Variable MCQs Question and Answer in Engineering Maths

31. Let f1(z) = z2 and f2(z) = $$\overline {\text{z}} $$ be two complex variable function. Here $$\overline {\text{z}} $$ is the complex conjugate of z. Choose the correct answer.

Answer & Solution

32. If $${\text{W}} = \phi + {\text{i}}\psi $$ represents the complex potential for an electric field. Given $$\psi = {{\text{x}}^2} - {{\text{y}}^2} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}},$$ then the function $$\phi $$ is

Answer & Solution

33. An integral $$I$$ over a counter-clockwise circle C is given by $$I = \oint_{\text{C}} {\frac{{{{\text{z}}^2} - 1}}{{{{\text{z}}^2} + 1}}{{\text{e}}^{\text{z}}}{\text{dz}}{\text{.}}} $$ If C is defined as |z| = 3, then the value of $$I$$ is

Answer & Solution

34. The function $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^2} + 4}}$$ is singular at

Answer & Solution

35. If f(z) has a pole of order n at z = a, then Residue of function f(z) at a is

Answer & Solution

36. Which one of the following functions is analytic over the entire complex plane?

Answer & Solution

37. Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

Answer & Solution

38. An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y), where $${\text{i}} = \sqrt { - 1} .$$ If u(x, y) = 2xy, then v(x, y) must be

Answer & Solution

39. Polar form of the Cauchy-Reimann equations is

Answer & Solution

40. 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) = zz∗ where z∗ denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane

Answer & Solution

31.
Let f₁(z) = z² and f₂(z) = $$\overline {\text{z}} $$ be two complex variable function. Here $$\overline {\text{z}} $$ is the complex conjugate of z. Choose the correct answer.

32.
If $${\text{W}} = \phi + {\text{i}}\psi $$ represents the complex potential for an electric field. Given $$\psi = {{\text{x}}^2} - {{\text{y}}^2} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}},$$ then the function $$\phi $$ is

33.
An integral $$I$$ over a counter-clockwise circle C is given by $$I = \oint_{\text{C}} {\frac{{{{\text{z}}^2} - 1}}{{{{\text{z}}^2} + 1}}{{\text{e}}^{\text{z}}}{\text{dz}}{\text{.}}} $$
If C is defined as |z| = 3, then the value of $$I$$ is

34.
The function $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^2} + 4}}$$ is singular at

35.
If f(z) has a pole of order n at z = a, then Residue of function f(z) at a is

36.
Which one of the following functions is analytic over the entire complex plane?

37.
Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

38.
An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y), where $${\text{i}} = \sqrt { - 1} .$$ If u(x, y) = 2xy, then v(x, y) must be

39.
Polar form of the Cauchy-Reimann equations is

40.
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) = zz^∗ where z^∗ denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane