Complex Variable MCQs Question and Answer in Engineering Maths Page-2 section-1

11.
If f(z) = (x² + ay²) + i bxy is a complex analytic function of z = x + iy, where $${\text{i}} = \sqrt { - 1} ,$$ then

A. a = -1, b = -1

B. a = -1, b = 2

C. a = 1, b = 2

D. a = 2, b = 2

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12.
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 = xy, the expression for v should be

A. $$\frac{{{{\left( {{\text{x}} + {\text{y}}} \right)}^2}}}{2} + {\text{k}}$$

B. $$\frac{{{{\text{x}}^2} - {{\text{y}}^2}}}{2} + {\text{k}}$$

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

D. $$\frac{{{{\left( {{\text{x}} - {\text{y}}} \right)}^2}}}{2} + {\text{k}}$$

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13.
F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i$$I$$m(z)
For what value of k will F(z) satisfy the Cauchy-Riemann equations

A. 0

B. 1

C. -1

D. y

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14.
The value of the integral $$\int\limits_{ - \infty }^\infty {\frac{{\sin {\text{x}}}}{{{{\text{x}}^2} + 2{\text{x}} + 2}}{\text{dx}}} $$ evaluated using contour integration and the residue theorem is

A. $$\frac{{ - \pi \sin \left( 1 \right)}}{{\text{e}}}$$

B. $$\frac{{ - \pi \cos \left( 1 \right)}}{{\text{e}}}$$

C. $$\frac{{\sin \left( 1 \right)}}{{\text{e}}}$$

D. $$\frac{{\cos \left( 1 \right)}}{{\text{e}}}$$

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15.
Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below

The value of $$I$$ is

A. $$\frac{1}{2}{\text{i}}$$

B. $$\frac{2}{3}{\text{i}}$$

C. $$\frac{3}{4}{\text{i}}$$

D. $$\frac{4}{5}{\text{i}}$$

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16.
The product of a complex number z = x + iy and its complex conjugate $$\overline {\text{z}} $$ is

A. x²

B. y²

C. x² - y²

D. x² + y²

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17.
If z = x + jy, where x and y are real, the value of |e^jz| is

A. 1

B. $${\text{e}}\sqrt {{{\text{x}}^2} + {{\text{y}}^2}} $$

C. e^y

D. e^-y

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18.
The value of $$\oint {\frac{1}{{{{\text{z}}^2}}}{\text{dz,}}} $$ where the contour is the unit circle traversed clockwise, is

A. -2πi

B. 0

C. 2πi

D. 4πi

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19.
Evaluate $$\oint_{\text{c}} {\frac{1}{{{{\left( {{\text{z}} - 1} \right)}^3} \cdot \left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ where c is the rectangular region defined by x = 0, x = 4, y = -1 and y = 1

A. 1

B. 0

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

D. $$\pi \left( {3 + 2{\text{i}}} \right)$$

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20.
If f(z) = C₀ + C₁z^-1, then $$\oint\limits_{{\text{unit circle}}} {\frac{{1 + {\text{f}}\left( {\text{z}} \right)}}{{\text{z}}}{\text{dz}}} $$ is given by

A. 2πC₁

B. 2π(1 + C₀)

C. 2πjC₁

D. 2πj(1 + C₀)

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Complex Variable MCQs Question and Answer in Engineering Maths

11. If f(z) = (x2 + ay2) + i bxy is a complex analytic function of z = x + iy, where $${\text{i}} = \sqrt { - 1} ,$$ then

Answer & Solution

12. 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 = xy, the expression for v should be

Answer & Solution

13. F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i$$I$$m(z) For what value of k will F(z) satisfy the Cauchy-Riemann equations

Answer & Solution

14. The value of the integral $$\int\limits_{ - \infty }^\infty {\frac{{\sin {\text{x}}}}{{{{\text{x}}^2} + 2{\text{x}} + 2}}{\text{dx}}} $$ evaluated using contour integration and the residue theorem is

Answer & Solution

15. Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below The value of $$I$$ is

Answer & Solution

16. The product of a complex number z = x + iy and its complex conjugate $$\overline {\text{z}} $$ is

Answer & Solution

17. If z = x + jy, where x and y are real, the value of |ejz| is

Answer & Solution

18. The value of $$\oint {\frac{1}{{{{\text{z}}^2}}}{\text{dz,}}} $$ where the contour is the unit circle traversed clockwise, is

Answer & Solution

19. Evaluate $$\oint_{\text{c}} {\frac{1}{{{{\left( {{\text{z}} - 1} \right)}^3} \cdot \left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ where c is the rectangular region defined by x = 0, x = 4, y = -1 and y = 1

Answer & Solution

20. If f(z) = C0 + C1z-1, then $$\oint\limits_{{\text{unit circle}}} {\frac{{1 + {\text{f}}\left( {\text{z}} \right)}}{{\text{z}}}{\text{dz}}} $$ is given by

Answer & Solution

11.
If f(z) = (x² + ay²) + i bxy is a complex analytic function of z = x + iy, where $${\text{i}} = \sqrt { - 1} ,$$ then

12.
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 = xy, the expression for v should be

13.
F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i$$I$$m(z)
For what value of k will F(z) satisfy the Cauchy-Riemann equations

14.
The value of the integral $$\int\limits_{ - \infty }^\infty {\frac{{\sin {\text{x}}}}{{{{\text{x}}^2} + 2{\text{x}} + 2}}{\text{dx}}} $$ evaluated using contour integration and the residue theorem is

15.
Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below

The value of $$I$$ is

16.
The product of a complex number z = x + iy and its complex conjugate $$\overline {\text{z}} $$ is

17.
If z = x + jy, where x and y are real, the value of |e^jz| is

18.
The value of $$\oint {\frac{1}{{{{\text{z}}^2}}}{\text{dz,}}} $$ where the contour is the unit circle traversed clockwise, is

19.
Evaluate $$\oint_{\text{c}} {\frac{1}{{{{\left( {{\text{z}} - 1} \right)}^3} \cdot \left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ where c is the rectangular region defined by x = 0, x = 4, y = -1 and y = 1

20.
If f(z) = C₀ + C₁z^-1, then $$\oint\limits_{{\text{unit circle}}} {\frac{{1 + {\text{f}}\left( {\text{z}} \right)}}{{\text{z}}}{\text{dz}}} $$ is given by