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 point z has been plotted in the complex plane, as shown in figure below.
Complex Variable mcq question image

The plot for point $$\frac{1}{{\text{z}}}$$ is

A

B

C

D

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

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

A

x² + y² + constant

B

x² - y² + constant

C

-x² + y² + constant

D

-x² - y² + constant

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
Complex Variable mcq question image

A

jπ

B

-jπ

C

-π

D

π

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

A

-x² + y² + constant

B

x² - y² + constant

C

x² + y² + constant

D

-(x² + y²) + constant

Contour C in the adjoining figure is described by x² + y² = 16. The value $$\oint\limits_{\text{c}} {\frac{{{{\text{z}}^2} + 8}}{{0.5{\text{z}} - 1.5{\text{j}}}}{\text{dz}}} $$ is (Note: $${\text{j}} = \sqrt { - 1} $$ )
Complex Variable mcq question image

A

-2πj

B

2πj

C

4πj

D

-4πj

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

$$\frac{4}{5}{\text{i}}$$

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) = x² - y², then expression for v(x, y) in terms of x, y and a general constant c would be

A

xy + c

B

$$\frac{{{{\text{x}}^2} + {{\text{y}}^2}}}{2} + {\text{c}}$$

C

2xy + c

D

$$\frac{{{{\left( {{\text{x}} - {\text{y}}} \right)}^2}}}{2} + {\text{c}}$$

If a function f(z) = u (x, y) + iv (x, y) of the complex variable z = x + iy, where x, y, u and v are real, is analytic in a domain D of z, then which of the following is true?

A

$$\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial x}}$$

B

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

C

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

D

$$\frac{{{\partial ^2}u}}{{\partial x\partial y}} = \frac{{{\partial ^2}v}}{{\partial x\partial y}}$$

Consider the following assertions.
1. Synthetic apirori judgment is the standard of knowledge.
2. Synthetic a posteriori judgement is the standard of knowledge.
3. Analytic apriori judgement is the standard of knowledge.
4. Analytic a posteriori judgement is the standard of knowledge.
Select the correct answer:

A

Only 1 is correct

B

Only 2 is correct

C

1 and 4 are correct

D

Only 3 is correct

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,

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