Complex Variable MCQs Question and Answer in Engineering Maths Page-7 section-1

61.
Given f(z) = g(z) + h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?

A. If f(z) is differentiable at z₀, then g(z) and h(z) are also differentiable at z₀

B. If g(z) and h(z) are differentiable at z₀, then f(z) is also differentiable at z₀

C. If f(z) is continuous at z₀, then it is differentiable at z₀

D. If f(z) is differentiable at z₀, then so are its real and imaginary parts

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62.
All the values of the multi-valued complex function 1ⁱ, where $${\text{i}} = \sqrt { - 1} ,$$ are

A. purely imaginary

B. real and non-negative

C. on the unit circle

D. equal in real and imaginary parts

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63.
If f(x + iy) = x³ - 3xy² + i$$\phi $$(x, y) where $${\text{i}} = \sqrt { - 1} $$ and f(x + iy) is an analytic function then $$\phi $$(x, y) is

A. y³ - 3x²y

B. 3x²y - y³

C. x⁴ - 4x²y

D. xy - y²

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64.
Potential function $$\phi $$ is given as $$\phi $$ = x² - y². What will be the stream function $$\psi $$ with the condition $$\psi $$ = 0 at x = y = 0?

A. 2xy

B. x² + y²

C. x² - y²

D. 2x²y²

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65.
Consider likely applicability of Cauchy's Integral Theorem to evaluate the following integral counter clockwise around the unit circle c.
$$I = \oint\limits_{\text{c}} {\sec {\text{z}}} {\text{dz,}}$$ z being a complex variable. The value of $$I$$ will be

A. $$I$$ = 0 : singularities set = $$\phi $$

B. $$I$$ = 0 : singularities set = $$\left\{ { \pm \frac{{2{\text{n}} + 1}}{2}\pi ;\,{\text{n}} = 0,\,1,\,2\,...} \right\}$$

C. $$I = \frac{\pi }{2}:$$ singularities set = $$\left\{ { \pm {\text{n}}\pi ;\,{\text{n}} = 0,\,1,\,2\,...} \right\}$$

D. None of above

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66.
For a complex number z = 1 - 4i with $${\text{i}} = \sqrt { - 1} ,$$ the value of $$\left| {\frac{{{\text{z}} + 3}}{{{\text{z}} - 1}}} \right|$$ is

A. 0

B. $$\frac{1}{{\sqrt 2 }}$$

C. 1

D. $$\sqrt 2 $$

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67.
The value of the contour integral $$\oint\limits_{\left| {{\text{z}} - {\text{i}}} \right| = 2} {\frac{1}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ in positive sense is

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

B. $$ - \frac{\pi }{2}$$

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

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

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68.
The value of $$\oint {\Gamma \frac{{3{\text{z}} - 5}}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}{\text{dz}}} $$ along a closed path $$\Gamma $$ is is equal to (4πi), where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The correct path $$\Gamma $$ is

A. Complex Variable mcq question image

B. Complex Variable mcq question image

C. Complex Variable mcq question image

D. Complex Variable mcq question image

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69.
The contour integral $$\oint\limits_{\text{C}} {{{\text{e}}^{\frac{1}{{\text{z}}}}}{\text{dz}}} $$ with C as the counter-clockwise unit circle in the z-plane is equal to

A. 0

B. $$2\pi $$

C. $$2\pi \sqrt { - 1} $$

D. $$\infty $$

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70.
A complex function f(z) = u(x, y) + iv(x, y) and its complex conjugate, f'(z) = u(x, y) - iv(x, y) are both analytic in the entire complex plane, where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The function f is then given by

A. f(z) = x + iy

B. f(z) = x² - y² + i2xy

C. f(z) = constant

D. f(z) = x² + y²

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Complex Variable MCQs Question and Answer in Engineering Maths

61. Given f(z) = g(z) + h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?

Answer & Solution

62. All the values of the multi-valued complex function 1i, where $${\text{i}} = \sqrt { - 1} ,$$ are

Answer & Solution

63. If f(x + iy) = x3 - 3xy2 + i$$\phi $$(x, y) where $${\text{i}} = \sqrt { - 1} $$ and f(x + iy) is an analytic function then $$\phi $$(x, y) is

Answer & Solution

64. Potential function $$\phi $$ is given as $$\phi $$ = x2 - y2. What will be the stream function $$\psi $$ with the condition $$\psi $$ = 0 at x = y = 0?

Answer & Solution

65. Consider likely applicability of Cauchy's Integral Theorem to evaluate the following integral counter clockwise around the unit circle c. $$I = \oint\limits_{\text{c}} {\sec {\text{z}}} {\text{dz,}}$$ z being a complex variable. The value of $$I$$ will be

Answer & Solution

66. For a complex number z = 1 - 4i with $${\text{i}} = \sqrt { - 1} ,$$ the value of $$\left| {\frac{{{\text{z}} + 3}}{{{\text{z}} - 1}}} \right|$$ is

Answer & Solution

67. The value of the contour integral $$\oint\limits_{\left| {{\text{z}} - {\text{i}}} \right| = 2} {\frac{1}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ in positive sense is

Answer & Solution

68. The value of $$\oint {\Gamma \frac{{3{\text{z}} - 5}}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}{\text{dz}}} $$ along a closed path $$\Gamma $$ is is equal to (4πi), where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The correct path $$\Gamma $$ is

Answer & Solution

69. The contour integral $$\oint\limits_{\text{C}} {{{\text{e}}^{\frac{1}{{\text{z}}}}}{\text{dz}}} $$ with C as the counter-clockwise unit circle in the z-plane is equal to

Answer & Solution

70. A complex function f(z) = u(x, y) + iv(x, y) and its complex conjugate, f'(z) = u(x, y) - iv(x, y) are both analytic in the entire complex plane, where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The function f is then given by

Answer & Solution

61.
Given f(z) = g(z) + h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?

62.
All the values of the multi-valued complex function 1ⁱ, where $${\text{i}} = \sqrt { - 1} ,$$ are

63.
If f(x + iy) = x³ - 3xy² + i$$\phi $$(x, y) where $${\text{i}} = \sqrt { - 1} $$ and f(x + iy) is an analytic function then $$\phi $$(x, y) is

64.
Potential function $$\phi $$ is given as $$\phi $$ = x² - y². What will be the stream function $$\psi $$ with the condition $$\psi $$ = 0 at x = y = 0?

65.
Consider likely applicability of Cauchy's Integral Theorem to evaluate the following integral counter clockwise around the unit circle c.
$$I = \oint\limits_{\text{c}} {\sec {\text{z}}} {\text{dz,}}$$ z being a complex variable. The value of $$I$$ will be

66.
For a complex number z = 1 - 4i with $${\text{i}} = \sqrt { - 1} ,$$ the value of $$\left| {\frac{{{\text{z}} + 3}}{{{\text{z}} - 1}}} \right|$$ is

67.
The value of the contour integral $$\oint\limits_{\left| {{\text{z}} - {\text{i}}} \right| = 2} {\frac{1}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ in positive sense is

68.
The value of $$\oint {\Gamma \frac{{3{\text{z}} - 5}}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}{\text{dz}}} $$ along a closed path $$\Gamma $$ is is equal to (4πi), where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The correct path $$\Gamma $$ is

69.
The contour integral $$\oint\limits_{\text{C}} {{{\text{e}}^{\frac{1}{{\text{z}}}}}{\text{dz}}} $$ with C as the counter-clockwise unit circle in the z-plane is equal to

70.
A complex function f(z) = u(x, y) + iv(x, y) and its complex conjugate, f'(z) = u(x, y) - iv(x, y) are both analytic in the entire complex plane, where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The function f is then given by