Complex Variable MCQs Question and Answer in Engineering Maths Page-9 section-1

81.
$$\oint {\frac{{{{\text{z}}^2} - 4}}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ evaluated anticlockwise around the circle |z - i| = 2, where $${\text{i}} = \sqrt { - 1} ,$$ is

A. -4π

B. 0

C. 2 + π

D. 2 + 2i

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82.
An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x, y) + iv(x, y). Then, u(x, y) and v(x, y) must satisfy,

A. $$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{x}}}}$$

B. $$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{x}}}}$$

C. $$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{x}}}}$$

D. $$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{x}}}}$$

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83.
In the neighborhood of z = 1, the function f(z) has a power series expansion of the form $${\text{f}}\left( {\text{z}} \right) = 1 + \left( {1 - {\text{z}}} \right) + {\left( {1 - {\text{z}}} \right)^2} + \,...$$
Then f(z) is

A. $$\frac{1}{{\text{z}}}$$

B. $$\frac{{ - 1}}{{{\text{z}} - 2}}$$

C. $$\frac{{{\text{z}} - 1}}{{{\text{z}} + 2}}$$

D. $$\frac{1}{{2{\text{z}} - 1}}$$

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84.
Solutions of Laplace equation having continuous second-order partial derivatives are called

A. biharmonic functions

B. harmonic functions

C. conjugate harmonic functions

D. error functions

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85.
Square roots of -i, where $${\text{i}} = \sqrt { - 1} ,$$ are

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

B. $${\text{cos}}\left( { - \frac{\pi }{4}} \right) + {\text{i}}\sin \left( { - \frac{\pi }{4}} \right),\,\cos \left( {\frac{{3\pi }}{4}} \right) + {\text{i}}\sin \left( {\frac{{3\pi }}{4}} \right)$$

C. $${\text{cos}}\left( {\frac{\pi }{4}} \right) + {\text{i}}\sin \left( {\frac{{3\pi }}{4}} \right),\,\cos \left( {\frac{{3\pi }}{4}} \right) + {\text{i}}\sin \left( {\frac{\pi }{4}} \right)$$

D. $${\text{cos}}\left( {\frac{{3\pi }}{4}} \right) + {\text{i}}\sin \left( { - \frac{{3\pi }}{4}} \right),\,\cos \left( { - \frac{{3\pi }}{4}} \right) + {\text{i}}\sin \left( {\frac{{3\pi }}{4}} \right)$$

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86.
Given two complex numbers $${{\text{z}}_1} = 5 + \left( {5\sqrt 3 } \right){\text{i}}$$ and $${{\text{z}}_2} = \frac{2}{{\sqrt 3 }} + 2{\text{i}}$$ the argument $$\frac{{{{\text{z}}_1}}}{{{{\text{z}}_2}}}$$ in degree is

A. 0

B. 30

C. 60

D. 90

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87.
The value of the contour integral in the complex plane $$\oint {\frac{{{{\text{z}}^3} - 2{\text{z}} + 3}}{{{\text{z}} - 2}}{\text{dz}}} $$ along the contour |z| = 3, taken counter-clockwise is

A. -18πi

B. 0

C. 14πi

D. 48πi

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88.
In the Laurent expansion of $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$ valid in the region 1 < | z | < 2, the coefficient of $$\frac{1}{{{{\text{z}}^2}}}$$ is

A. 0

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

C. 1

D. -1

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89.
If $${\text{x}} = \sqrt { - 1} ,$$ then the value of x^x is

A. $${{\text{e}}^{ - \frac{\pi }{2}}}$$

B. $${{\text{e}}^{\frac{\pi }{2}}}$$

C. x

D. 1

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90.
For an analytic function, f(x + iy) = u(i, y) + iv(i, y), u is given by u = 3x² - 3y. The expression for v, considering K to be a constant is

A. 3y² - 3x² + K

B. 6x - 6y + K

C. 6y - 6x + K

D. 6xy + K

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Complex Variable MCQs Question and Answer in Engineering Maths

81. $$\oint {\frac{{{{\text{z}}^2} - 4}}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ evaluated anticlockwise around the circle |z - i| = 2, where $${\text{i}} = \sqrt { - 1} ,$$ is

Answer & Solution

82. An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x, y) + iv(x, y). Then, u(x, y) and v(x, y) must satisfy,

Answer & Solution

83. In the neighborhood of z = 1, the function f(z) has a power series expansion of the form $${\text{f}}\left( {\text{z}} \right) = 1 + \left( {1 - {\text{z}}} \right) + {\left( {1 - {\text{z}}} \right)^2} + \,...$$ Then f(z) is

Answer & Solution

84. Solutions of Laplace equation having continuous second-order partial derivatives are called

Answer & Solution

85. Square roots of -i, where $${\text{i}} = \sqrt { - 1} ,$$ are

Answer & Solution

86. Given two complex numbers $${{\text{z}}_1} = 5 + \left( {5\sqrt 3 } \right){\text{i}}$$ and $${{\text{z}}_2} = \frac{2}{{\sqrt 3 }} + 2{\text{i}}$$ the argument $$\frac{{{{\text{z}}_1}}}{{{{\text{z}}_2}}}$$ in degree is

Answer & Solution

87. The value of the contour integral in the complex plane $$\oint {\frac{{{{\text{z}}^3} - 2{\text{z}} + 3}}{{{\text{z}} - 2}}{\text{dz}}} $$ along the contour |z| = 3, taken counter-clockwise is

Answer & Solution

88. In the Laurent expansion of $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$ valid in the region 1 < | z | < 2, the coefficient of $$\frac{1}{{{{\text{z}}^2}}}$$ is

Answer & Solution

89. If $${\text{x}} = \sqrt { - 1} ,$$ then the value of xx is

Answer & Solution

90. For an analytic function, f(x + iy) = u(i, y) + iv(i, y), u is given by u = 3x2 - 3y. The expression for v, considering K to be a constant is

Answer & Solution

81.
$$\oint {\frac{{{{\text{z}}^2} - 4}}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ evaluated anticlockwise around the circle |z - i| = 2, where $${\text{i}} = \sqrt { - 1} ,$$ is

82.
An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x, y) + iv(x, y). Then, u(x, y) and v(x, y) must satisfy,

83.
In the neighborhood of z = 1, the function f(z) has a power series expansion of the form $${\text{f}}\left( {\text{z}} \right) = 1 + \left( {1 - {\text{z}}} \right) + {\left( {1 - {\text{z}}} \right)^2} + \,...$$
Then f(z) is

84.
Solutions of Laplace equation having continuous second-order partial derivatives are called

85.
Square roots of -i, where $${\text{i}} = \sqrt { - 1} ,$$ are

86.
Given two complex numbers $${{\text{z}}_1} = 5 + \left( {5\sqrt 3 } \right){\text{i}}$$ and $${{\text{z}}_2} = \frac{2}{{\sqrt 3 }} + 2{\text{i}}$$ the argument $$\frac{{{{\text{z}}_1}}}{{{{\text{z}}_2}}}$$ in degree is

87.
The value of the contour integral in the complex plane $$\oint {\frac{{{{\text{z}}^3} - 2{\text{z}} + 3}}{{{\text{z}} - 2}}{\text{dz}}} $$ along the contour |z| = 3, taken counter-clockwise is

88.
In the Laurent expansion of $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$ valid in the region 1 < | z | < 2, the coefficient of $$\frac{1}{{{{\text{z}}^2}}}$$ is

89.
If $${\text{x}} = \sqrt { - 1} ,$$ then the value of x^x is

90.
For an analytic function, f(x + iy) = u(i, y) + iv(i, y), u is given by u = 3x² - 3y. The expression for v, considering K to be a constant is