The integral $$\oint {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ evaluated around the unit circle on the complex plane for $${\text{f}}\left( {\text{z}} \right) = \frac{{\cos {\text{z}}}}{{\text{z}}}$$ is

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

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

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

A common convention is: • is evaluated first • and are evaluated next • Quantifiers are evaluated next • is evaluated last.

A

True

B

False

Statement : It has been reported by one of the TV channels that the answer papers of Board examination of one State are evaluated by students studying in the same standard with the help of model answers as instructed by the examiners.

Courses of Action :
I. All such examiners should be immediately suspended from their official positions.
II. All such papers evaluated by the students should be immediately confiscated and got evaluated by qualified teachers.
III. The Board should explore possibilities even though they are remote, of getting the answer papers of this examination evaluated by computerised machines.

A

Only I follows

B

Only II follows

C

Only III follows

D

Only I and II follow

E

All follow

A common convention is:
• is evaluated first
• and are evaluated next
• Quantifiers are evaluated next
• is evaluated last.

A

True

B

False

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

The contour integral $$\oint {\frac{{dz}}{{{z^2} + {a^2}}}} $$ is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points

A

$$\frac{{1 + i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 + i}}{{\sqrt 2 }}a$$

B

$$ia{\text{ and }} - ia$$

C

$$ia,\, - ia,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$

D

$$\frac{{1 + i}}{{\sqrt 2 }}a,\, - \frac{{1 + i}}{{\sqrt 2 }}a,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
$$\iint\limits_{\text{S}} {\frac{1}{4}\left( {{\text{F}} \cdot {\text{n}}} \right){\text{dA}}}$$ where S is the sphere, x² + y² + z² = 1 and n is the outward unit normal vector to the sphere.
The value of the surface integral is

A

$$\pi $$

B

$$2\pi $$

C

$$\frac{{3\pi }}{4}$$

D

$$4\pi $$

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

$$\infty $$

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 integral $$\oint {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ evaluated around the unit circle on the complex plane for $${\text{f}}\left( {\text{z}} \right) = \frac{{\cos {\text{z}}}}{{\text{z}}}$$ is

Related Questions