Complex Variable — Part 2 MCQs with Explanations

$${2^{{\text{n}} + 1}}\cos \frac{{{\text{n}}\pi }}{6}$$ is the correct answer to “If n is a positive integer then, (√3 + i) n + (√3 - i) n is”. This question is standard in competitive examination syllabi for Complex Variable.

πi is the correct answer to “Integration of the complex function $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2}}}{{{{\text{z}}^2} - 1}},$$ in the counterclockwise direction, around |z - 1| = 1, is”. This question is standard in competitive examination syllabi for Complex Variable.

-0.511 + 1.57i is the correct answer to “If z is a complex variable, the value of $$\int\limits_5^{3{\text{i}}} {\frac{{{\text{dz}}}}{{\text{z}}}} $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

a pole of order 2 is the correct answer to “For the function $$\frac{{\sin {\text{z}}}}{{{{\text{z}}^3}}}$$ of a complex variable the point z = 0 is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{2}{5}$$ is the correct answer to “Let z be a complex variable. For a counterclockwise integration around a unit circle C, centered at origin. $$\oint_{\text{c}} {\frac{1}{{5{\text{z}} - 4}}{\text{dz}}} = {\text{A}}\pi {\text{i}}$$ the value of A is”. This question is standard in competitive examination syllabi for Complex Variable.

$${{\text{e}}^{ - \frac{\pi }{2}}}$$ is the correct answer to “Let $${\text{j}} = \sqrt { - 1} .$$ Then one value of j j is”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “The sum of residues of $${\text{f}}\left( {\text{z}} \right) = \frac{{2{\text{z}}}}{{{{\left( {{\text{z}} - 1} \right)}^2}\left( {{\text{z}} - 2} \right)}}$$ at its singular point is”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “The value of the Integral $$\oint\limits_{\text{c}} {\frac{{ - 3{\text{z}} + 4}}{{\left( {{{\text{z}}^2} + 4{\text{z}} + 5} \right)}}{\text{dz}}} $$ where c is the circle |z| = 1 is given by”. This question is standard in competitive examination syllabi for Complex Variable.

π is the correct answer to “The value of $$\oint\limits_{\text{C}} {\frac{{{\text{dz}}}}{{\left( {1 + {{\text{z}}^2}} \right)}}} $$ where C is the contour |z - i/2| = 1 is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{\pi {\text{i}}}}{2}$$ is the correct answer to “The value of the integral $$\oint {\frac{{{\text{z}} + 1}}{{{{\text{z}}^2} - 4}}{\text{dz}}} $$ in counter clockwise direction around a circle C of radius 1 with center at the point z = -2 is”. This question is standard in competitive examination syllabi for Complex Variable.

2πi is the correct answer to “The integral $$\oint {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ evaluated around the unit circle on the complex plane for $${\text{f}}\left( {\text{z}} \right) = \frac{{\cos {\text{z}}}}{{\text{z}}}$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{1}{{125}}{\text{ and }}\frac{{ - 1}}{{125}}$$ is the correct answer to “The residues of a function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 4} \right){{\left( {{\text{z}} + 1} \right)}^3}}}$$ are”. This question is standard in competitive examination syllabi for Complex Variable.

1 + 2i is the correct answer to “The value of expression $$\frac{{ - 5 + {\text{i}}10}}{{3 + 4{\text{i}}}}$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{\pi }{2}$$ is the correct answer to “The argument of the complex number $$\frac{{1 + {\text{i}}}}{{1 - {\text{i}}}},$$ where $${\text{i}} = \sqrt { - 1} ,$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

Simple pole is the correct answer to “The nature of singularity of function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\cos {\text{z}} - \sin {\text{z}}}}$$ at $${\text{z}} = \frac{\pi }{4}$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

i and -i is the correct answer to “The analytic function $${\text{f}}\left( {\text{z}} \right) = \frac{{{\text{z}} - 1}}{{{{\text{z}}^2} + 1}}$$ has singularities at”. This question is standard in competitive examination syllabi for Complex Variable.

i. 7.39, ii. 0 is the correct answer to “The values of the integral $$\frac{1}{{2\pi {\text{j}}}}\oint\limits_{\text{c}} {\frac{{{{\text{e}}^{\text{z}}}}}{{{\text{z}} - 2}}{\text{dz}}} $$ along a closed contour c in anti-clockwise direction for i. the point z 0 = 2 inside the contour c, and ii. the point z 0 = 2 outside the contour c, respectively, are”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “Evaluate: $$\int\limits_{\text{c}} {\frac{{{\text{dz}}}}{{{\text{z}}\sin {\text{z}}}}} ,$$ where c is x 2 + y 2 = 1.”. This question is standard in competitive examination syllabi for Complex Variable.

If g(z) and h(z) are differentiable at z0, then f(z) is also differentiable at z0 is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

real and non-negative is the correct answer to “All the values of the multi-valued complex function 1 i , where $${\text{i}} = \sqrt { - 1} ,$$ are”. This question is standard in competitive examination syllabi for Complex Variable.

3x2y - y3 is the correct answer to “If f(x + iy) = x 3 - 3xy 2 + i$$\phi $$(x, y) where $${\text{i}} = \sqrt { - 1} $$ and f(x + iy) is an analytic function then $$\phi $$(x, y) is”. This question is standard in competitive examination syllabi for Complex Variable.

2xy is the correct answer to “Potential function $$\phi $$ is given as $$\phi $$ = x 2 - y 2 . What will be the stream function $$\psi $$ with the condition $$\psi $$ = 0 at x = y = 0”. This question is standard in competitive examination syllabi for Complex Variable.

$$I$$ = 0 : singularities set = $$\phi $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

$$\sqrt 2 $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{\pi }{2}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

$$2\pi \sqrt { - 1} $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

f(z) = constant is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{1}{2}\left( {\cos \frac{\pi }{2} + {\text{i}}\sin \frac{\pi }{2}} \right)$$ is the correct answer to “The complex number $${\left( {\frac{{2 + {\text{i}}}}{{3 - {\text{i}}}}} \right)^2}$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{1}{2},\,1{\text{ and }} - \frac{3}{2}$$ is the correct answer to “The residues of a complex function $${\text{X}}\left( {\text{z}} \right) = \frac{{1 - 2{\text{z}}}}{{{\text{z}}\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$ at its poles are”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{{{\text{z}}^2} - 1}}{{{\text{z}} + 2}}$$ is the correct answer to “Which one of the following functions is analytic in the region |z| ≤ 1”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{48\pi {\text{i}}}}{{13}}$$ is the correct answer to “The value of the integral $$\oint {\frac{{2{\text{z}} + 5}}{{\left( {{\text{z}} - \frac{1}{2}} \right)\left( {{{\text{z}}^2} - 4{\text{z}} + 5} \right)}}{\text{dz}}} $$ over the contour |z| = 1, taken in the anti-clockwise direction, would be”. This question is standard in competitive examination syllabi for Complex Variable.

2xy + c is the correct answer to “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) = x 2 - y 2 , then expression for v(x, y) in terms of x, y and a general constant c would be”. This question is standard in competitive examination syllabi for Complex Variable.

$$ - \frac{1}{{32}}$$ is the correct answer to “The residue of the function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{{{\left( {{\text{z}} + 2} \right)}^2}{{\left( {{\text{z}} - 2} \right)}^2}}}$$ at z = 2 is”. This question is standard in competitive examination syllabi for Complex Variable.

2i is the correct answer to “For a complex number z, $$\mathop {\lim }\limits_{{\text{z}} \to {\text{i}}} \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^3} + 2{\text{z}} - {\text{i}}\left( {{{\text{z}}^2} + 2} \right)}}$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{27}}{{16}}$$ is the correct answer to “The residue of $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^3}}}{{{{\left( {{\text{z}} - 1} \right)}^4}\left( {{\text{z}} - 2} \right)\left( {{\text{z}} - 3} \right)}}$$ at z = 3 is”. This question is standard in competitive examination syllabi for Complex Variable.

1 is the correct answer to “Given $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{{\text{z}} + 1}} - \frac{2}{{{\text{z}} + 3}}.$$ If C is a counter clockwise path in the z-plane such that |z + 1| = 1, the value of $$\frac{1}{{2\pi {\text{j}}}}\oint_{\text{C}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “If a complex number ω satisfied the equation ω 3 = 1 then value of $$1 + \omega + \frac{1}{\omega }$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

-4π is the correct answer to “$$\oint {\frac{{{{\text{z}}^2} - 4}}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ evaluated anticlockwise around the circle |z - i| = 2, where $${\text{i}} = \sqrt { - 1} ,$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{v}}}}{{\partial {\text{y}}}}{\text{ and }}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = \frac{{ - \partial {\text{v}}}}{{\partial {\text{x}}}}$$ is the correct answer to “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,”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{1}{{\text{z}}}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.