Complex Variable — Part 1 MCQs with Explanations

-x2 + y2 + constant is the correct answer to “f(z) = u(x, y) + iv(x, y) is an analytic function or complex variable z = x + iy where $${\text{i}} = \sqrt { - 1} ,$$ u(x, y) = 2xy, then v(x, y) may be expressed as”. This question is standard in competitive examination syllabi for Complex Variable.

17 + 6i is the correct answer to “The product of complex numbers (3 - 2i) and (3 + i4) results in”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{ - 1}}{2} + \frac{{{\text{i}}{{\sqrt 3 }^2}}}{2}$$ is the correct answer to “If a complex number $${\text{z}} = \frac{{\sqrt 3 }}{2} + {\text{i}}\frac{1}{2}$$ then z 4 is”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “If C is a circle of radius r with centre z 0 , in the complex z-plane and if n is a non-zero integer, then $$\oint {\frac{{{\text{dz}}}}{{{{\left( {{\text{z}} - {{\text{z}}_0}} \right)}^{{\text{n}} + 1}}}}} $$ equals”. This question is standard in competitive examination syllabi for Complex Variable.

-4πj is the correct answer to “Contour C in the adjoining figure is described by x 2 + y 2 = 16. The value $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^2} + 8}}{{0.5{\text{z}} - 1.5{\text{j}}}}{\text{dz}}} $$ is (Note: $${\text{j}} = \sqrt { - 1} $$ )”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “The value of $$\oint {\frac{{\sin {\text{z}}}}{{\text{z}}}{\text{dz,}}} $$ where the counter of integration is a simple closed curve around the origin, is”. This question is standard in competitive examination syllabi for Complex Variable.

-0.5 + 0.5i is the correct answer to “$${\text{z}} = \frac{{2 - 3{\text{j}}}}{{ - 5 + {\text{i}}}}$$ can be expressed as”. This question is standard in competitive examination syllabi for Complex Variable.

-1 is the correct answer to “Consider the following complex function $${\text{f}}\left( {\text{z}} \right) = \frac{9}{{\left( {{\text{z}} - 1} \right){{\left( {{\text{z}} + 2} \right)}^2}}}$$ Which of the following is one of the residues of the above function”. This question is standard in competitive examination syllabi for Complex Variable.

$$ - \pi \frac{{\text{i}}}{2}$$ is the correct answer to “The value of $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^2}}}{{{{\text{z}}^4} - 1}}{\text{dz}}} $$ using Cauchy's integral formula, around circle |z + 1| = 1 where z = x + iy is”. This question is standard in competitive examination syllabi for Complex Variable.

√5 is the correct answer to “The modulus of the complex number $$\left( {\frac{{3 + 4{\text{i}}}}{{1 - 2{\text{i}}}}} \right)$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

a = -1, b = 2 is the correct answer to “If f(z) = (x 2 + ay 2 ) + i bxy is a complex analytic function of z = x + iy, where $${\text{i}} = \sqrt { - 1} ,$$ then”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{{{\text{y}}^2} - {{\text{x}}^2}}}{2} + {\text{k}}$$ 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 = xy, the expression for v should be”. This question is standard in competitive examination syllabi for Complex Variable.

1 is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{ - \pi \sin \left( 1 \right)}}{{\text{e}}}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{2}{3}{\text{i}}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

x2 + y2 is the correct answer to “The product of a complex number z = x + iy and its complex conjugate $$\overline {\text{z}} $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

e-y is the correct answer to “If z = x + jy, where x and y are real, the value of |e jz | is”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “The value of $$\oint {\frac{1}{{{{\text{z}}^2}}}{\text{dz,}}} $$ where the contour is the unit circle traversed clockwise, is”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “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”. This question is standard in competitive examination syllabi for Complex Variable.

2πj(1 + C0) is the correct answer to “If f(z) = C 0 + C 1 z -1 , then $$\oint\limits_{{\text{unit circle}}} {\frac{{1 + {\text{f}}\left( {\text{z}} \right)}}{{\text{z}}}{\text{dz}}} $$ is given by”. This question is standard in competitive examination syllabi for Complex Variable.

0 is the correct answer to “If C is a circle |z| = 4 and $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2}}}{{{{\left( {{{\text{z}}^2} - 3{\text{z}} + 2} \right)}^2}}},$$ then $$\oint\limits_{\text{c}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$ - \frac{4}{3}$$ is the correct answer to “What is the residue of the function $$\frac{{1 - {{\text{e}}^{2{\text{z}}}}}}{{{{\text{z}}^4}}}$$ at its pole”. This question is standard in competitive examination syllabi for Complex Variable.

cos $$\overline {\text{z}} $$ is the correct answer to “Let z = x + jy where $${\text{j}} = \sqrt { - 1} .$$ Then $$\overline {\cos {\text{z}}} = ?$$”. This question is standard in competitive examination syllabi for Complex Variable.

f(z) is continuous but not analytic is the correct answer to “Consider the function f(z) = z + z ∗ where z is a complex variable and z ∗ denotes its complex conjugate. Which one of the following is TRUE”. This question is standard in competitive examination syllabi for Complex Variable.

2x + i2y is the correct answer to “Consider the analytic function f(z) = x 2 - y 2 + i2xy of the complex variable z = x + iy, where $${\text{i}} = \sqrt { - 1} .$$ The derivative f'(z) is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{2\pi }}{{81}} - 4\pi {\text{i}}$$ is the correct answer to “Using Cauchy's integral theorem, the value of the integral (integration being taken in counterclockwise direction) $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^3} - 6}}{{3{\text{z}} - {\text{i}}}}{\text{dz}}} $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{2\pi {\text{i}}}}{5}$$ is the correct answer to “The value of the integral $$\int\limits_{\text{c}} {\frac{{\cos \left( {2\pi {\text{z}}} \right)}}{{\left( {2{\text{z}} - 1} \right)\left( {{\text{z}} - 3} \right)}}{\text{dz}}} $$ (where C is a closed curve given by |z| = 1) is”. This question is standard in competitive examination syllabi for Complex Variable.

jπ is the correct answer to “If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $$\oint\limits_{\text{D}} {\frac{1}{{\left( {{{\text{s}}^2} - 1} \right)}}{\text{ds}}} $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

4xy - 2x2 + 2y2 + constant is the correct answer to “A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x 2 - 2y 2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is”. This question is standard in competitive examination syllabi for Complex Variable.

$$I{\text{m}}\left( {\text{s}} \right) \ne \frac{{\left( {2{\text{n}} + 1} \right)\pi }}{2}$$ is the correct answer to “The complex function tan h(s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n”. This question is standard in competitive examination syllabi for Complex Variable.

Only f1(z) is analytic is the correct answer to “Let f 1 (z) = z 2 and f 2 (z) = $$\overline {\text{z}} $$ be two complex variable function. Here $$\overline {\text{z}} $$ is the complex conjugate of z. Choose the correct answer.”. This question is standard in competitive examination syllabi for Complex Variable.

$$ - 2{\text{xy}} + \frac{{\text{y}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$ is the correct answer to “If $${\text{W}} = \phi + {\text{i}}\psi $$ represents the complex potential for an electric field. Given $$\psi = {{\text{x}}^2} - {{\text{y}}^2} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}},$$ then the function $$\phi $$ is”. This question is standard in competitive examination syllabi for Complex Variable.

-4πi sin(1) is the correct answer to “An integral $$I$$ over a counter-clockwise circle C is given by $$I = \oint_{\text{C}} {\frac{{{{\text{z}}^2} - 1}}{{{{\text{z}}^2} + 1}}{{\text{e}}^{\text{z}}}{\text{dz}}{\text{.}}} $$ If C is defined as |z| = 3, then the value of $$I$$ is”. This question is standard in competitive examination syllabi for Complex Variable.

z = ±2i is the correct answer to “The function $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^2} + 4}}$$ is singular at”. This question is standard in competitive examination syllabi for Complex Variable.

$${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {{\text{n}} - 1} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{\text{n}}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$ is the correct answer to “If f(z) has a pole of order n at z = a, then Residue of function f(z) at a is”. This question is standard in competitive examination syllabi for Complex Variable.

cos(z) is the correct answer to “Which one of the following functions is analytic over the entire complex plane”. This question is standard in competitive examination syllabi for Complex Variable.

$$\overline {\text{z}} $$ (complex conjugate of z) is analytical function is the correct answer to “Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE”. This question is standard in competitive examination syllabi for Complex Variable.

-x2 + y2 + constant 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) = 2xy, then v(x, y) must be”. This question is standard in competitive examination syllabi for Complex Variable.

$$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = \frac{1}{{\text{r}}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - \frac{1}{{\text{r}}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$ is the correct answer to “Polar form of the Cauchy-Reimann equations is”. This question is standard in competitive examination syllabi for Complex Variable.

the point (1, 0) is the correct answer to “Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S = {z : |z| = 1}. Consider the function f(z) = zz ∗ where z ∗ denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane”. This question is standard in competitive examination syllabi for Complex Variable.