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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<br><img src="/images/question-image/engineering-maths/complex-variable/1689240191-consider-the-line-integral-texti-inttextc-left-textx-pow-2-textitexty-pow-2-righttextdz.png" title="Complex Variable mcq question image" alt="Complex Variable mcq question image"><br>The value of $$I$$ is
A
$$\frac{1}{2}{\text{i}}$$
B
$$\frac{2}{3}{\text{i}}$$
C
$$\frac{3}{4}{\text{i}}$$
D
$$\frac{4}{5}{\text{i}}$$
Correct Answer:
$$\frac{2}{3}{\text{i}}$$
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
A
jπ
B
-jπ
C
-π
D
π
The line integral $$\int\limits_{{{\text{P}}_1}}^{{{\text{P}}_2}} {\left( {{\text{ydx}} + {\text{xdy}}} \right)} $$ for P
1
(x
1
, y
1
) to P
2
(x
2
, y
2
) along the semicircle P
1
, P
2
shown in the figure is
A
x<sub>2</sub>y<sub>2</sub> - x<sub>1</sub>y<sub>1</sub>
B
$$\left( {{\text{y}}_2^2 - {\text{y}}_1^2} \right) + \left( {{\text{x}}_2^2 - {\text{x}}_1^2} \right)$$
C
(x<sub>2</sub> - x<sub>1</sub>) (y<sub>2</sub> - y<sub>1</sub>)
D
(y<sub>2</sub> - y<sub>1</sub>)<sup>2</sup> + (x<sub>2</sub> - x<sub>1</sub>)<sup>2</sup>
Consider the system shown in the figure below. The transfer function $$\frac{{Y\left( z \right)}}{{X\left( z \right)}}$$ of the system is
A
$$\frac{{1 + a{z^{ - 1}}}}{{1 + b{z^{ - 1}}}}$$
B
$$\frac{{1 + b{z^{ - 1}}}}{{1 + a{z^{ - 1}}}}$$
C
$$\frac{{1 + a{z^{ - 1}}}}{{1 - b{z^{ - 1}}}}$$
D
$$\frac{{1 - b{z^{ - 1}}}}{{1 - a{z^{ - 1}}}}$$
If a + b + c + d = 4, then find the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$ + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$ + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$ + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$ is?
A
0
B
5
C
1
D
4
If a + b + c + d = 4, then the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$ + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$ + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$ + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$ is?
A
0
B
1
C
4
D
1 + abcd
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?
A
1
B
2
C
3
D
4
Find out the alternative figure which contains figure (X) as its part.
A
1
B
2
C
3
D
4
Find out the alternative figure which contains figure (X) as its part.
A
1
B
2
C
3
D
4
Find out the alternative figure which contains figure (X) as its part.
A
1
B
2
C
3
D
4
Find out the alternative figure which contains figure (X) as its part.
A
1
B
2
C
3
D
4
