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

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

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

A

$$\frac{{2\pi }}{{81}} - 4\pi {\text{i}}$$

B

$$\frac{\pi }{8} - 6\pi {\text{i}}$$

C

$$\frac{{4\pi }}{{81}} - 6\pi {\text{i}}$$

D

1

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 z₀ = 2 inside the contour c, and
ii. the point z₀ = 2 outside the contour c, respectively, are

A

i. 2.72, ii. 0

B

i. 7.39, ii. 0

C

i. 0, ii. 2.72

D

i. 0, ii. 7.39

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

Considered the following statements about the characteristics of contours - (1) Closed contour lines with higher values inside show a lake, (2) Contour is an imaginary line joining points of equal elevations, (3) Closely spaced contour lines indicate a steep slope. (4) Contour lines of equal height may not converge nor continue as one line but a single contour line may split into two lines Which of the statements given above are correct?

A

2, 3 and 4

B

2 and 3

C

1 and 4

D

1 and 2

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

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

A

$$\frac{{24\pi {\text{i}}}}{{13}}$$

B

$$\frac{{48\pi {\text{i}}}}{{13}}$$

C

$$\frac{{24}}{{13}}$$

D

$$\frac{{12}}{{13}}$$

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

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

$$\infty $$

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

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

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

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