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

A) a = -1, b = -1 B) a = -1, b = 2 C) a = 1, b = 2 D) a = 2, b = 2

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Answer & Solution

Answer: Option B

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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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Answer: Option C

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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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Answer: Option B

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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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Answer: Option A

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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
Complex Variable mcq question image

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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Answer: Option B

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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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Answer: Option D

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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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Answer: Option D

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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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Answer: Option B

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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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Answer: Option B

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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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Answer: Option D

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