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<br><img src="/images/question-image/engineering-maths/complex-variable/1689240490-if-the-semi-circular-contour-0-of-radius-2-is-as-shown.png" title="Complex Variable mcq question image" alt="Complex Variable mcq question image">

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

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

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

The line integral $$\int\limits_{{{\text{P}}_1}}^{{{\text{P}}_2}} {\left( {{\text{ydx}} + {\text{xdy}}} \right)} $$ for P₁(x₁, y₁) to P₂(x₂, y₂) along the semicircle P₁, P₂ shown in the figure is
Calculus mcq question image

A

x₂y₂ - x₁y₁

B

$$\left( {{\text{y}}_2^2 - {\text{y}}_1^2} \right) + \left( {{\text{x}}_2^2 - {\text{x}}_1^2} \right)$$

C

(x₂ - x₁) (y₂ - y₁)

D

(y₂ - y₁)² + (x₂ - x₁)²

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

The pictorial view of the frustum of a square pyramid is shown in figure 'X'. Its top view, when viewed in the direction of the arrow, will look like which of the given alternatives 1, 2, 3 and 4?
Figure Formation and Analysis mcq question image

Figure Formation and Analysis mcq question image

A

1

B

2

C

3

D

4

Find out the alternative figure which contains figure (X) as its part.
Spotting Out The Embedded Figure mcq question image

A

1

B

2

C

3

D

4

Find out the alternative figure which contains figure (X) as its part.
Spotting Out The Embedded Figure mcq question image

A

1

B

2

C

3

D

4

Find out the alternative figure which contains figure (X) as its part.
Spotting Out The Embedded Figure mcq question image

A

1

B

2

C

3

D

4

Find out the alternative figure which contains figure (X) as its part.
Spotting Out The Embedded Figure mcq question image

A

1

B

2

C

3

D

4

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

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