Engineering Mathematics

Q. Consider the line integral $I={\int }_c(x^2+i{y}^2)dz$, where $z=x+iy$. The line $c$ is shown in the figure below



The value of $I$ is

1. $\frac{1}{2}i$
2. $\frac{2}{3}i$
3. $\frac{3}{4}i$
4. $\frac{4}{5}i$

Q. Let $z=x+iy$ be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

1. The residue of $\frac{z}{z^2-1}$ at $z=1$ is $1/2$
2. ${\oint }_C{z}^{2}dz=0$
3. $\frac{1}{2\pi i}{\oint }_C\frac{1}{z}dz=1$
4. $\overline{z}$ (complex conjugate of $z$) is an analytical function

Q. The value of the integral ${\oint }_C\frac{z+1}{z^2-4}dz$ in counter clockwise direction around a circle $C$ of radius $1$ with center at the point $z=-2$

1. $\frac{\pi i}{2}$
2. $2\pi i$
3. $-\frac{\pi i}{2}$
4. $-2\pi i$

Q. If $z$ is a complex variable, the value of ${\int }_5^{3i}\frac{dz}{z}$ is

1. $-0.511-1.57i$
2. $-0.511+1.57i$
3. $0.511-1.57i$
4. $0.511+1.57i$

Q. $C$ is a closed path in the $z-$plane given by $|z|=3$. The value of the integral ${\oint }_C\left(\frac{z^2-z+4j}{z+2j}\right)dz$ is

1. $-4\pi (1+j2)$
2. $4\pi (3+j2)$
3. $-4\pi (3+j2)$
4. $4\pi (1-j2)$

Q. An integra $I$ over a counterclockwise circle $C$ is given by $I={\oint }_C\frac{z^2-1}{z^2+1}{e}^{z}dz$
If $C$ is defined as $|z|=3$, then the value of $I$ is

1. $-\pi isin\left(1\right)$
2. $-2\pi isin\left(1\right)$
3. $-3\pi isin\left(1\right)$
4. $-4\pi isin\left(1\right)$

Q. Evaluate ${\int }_c\frac{1}{(z-1{)}^3.(z-3)}dz$
where $c$ is the rectangular region defined by $z=0,x=4,y=-1$ and $y=1$

1. $1$
2. $0$
3. $\frac{\pi }{2}i$
4. $\pi (3+2i)$

Q.

In the following integral, th contour $C$ encloses the points $2\pi j$ and $-2\pi j$.
$-\frac{1}{2\pi }{\oint }_C\frac{sinz}{(z-2\pi j{)}^3}dz$
The value of the integral is

1. 133.87

Q.

If $C$ denotes the counterclockwise unit circle, the value of the contour integral $\frac{1}{2\pi j}{\oint }_{C}Rezdz$ is

1. 0.5

Q. The countour $C$ in the adjoining figure is described by $x^2+y^2=16$. Then the value of ${\int }_c\frac{z^2+8}{\left(0.5\right)z-\left(1.5\right)j}dz$

1. $-2\pi j$
2. $-2\pi j$
3. $4\pi j$
4. $-4\pi j$