Residue Theorem

Q. The value of the integral ${\int }_c\frac{cos2\pi z}{(2z-1)(z-3)}dz$ (Where $C$ is a closed curve given by $|z|=1$ is

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

Q. The value of the contour integral in th complex-plane $\oint \frac{z^3-2z+3}{z-2}dz$ along the contour $|z|=3$, taken counter-clockwise is:

1. $18\pi i$
2. $0$
3. $14\pi i$
4. $48\pi i$

Q.

For the curve $y^n=a^{n-1}x$, the subnormal at any point is constant.

The value of n must be

1. $2$

2. $3$

3. $0$

4. $1$

Q. Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$. centred at origin.
$I={\oint }_c\frac{1}{5z-4}dz=A\pi i$
the value of $A$ is

1. $2/5$
2. $1/2$
3. $2$
4. $4/5$

Q. The value of the contour integral ${\oint }_{|z-j|=2}\frac{1}{z^2+4}dz$ in positive sense is

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

Q. The value of ${\oint }_C\frac{dz}{(1+z^2)}$ where $C$ is the contour $|z-\frac{1}{2}|=1$ is

1. $2\pi$
2. $\pi$
3. $ta{n}^{-1}z$
4. $\pi ita{n}^{-1}z$

Q. The value of the integral ${\int }_C\frac{2z+5}{(z-\frac{1}{2})(z^2-4z+5)}$
over the contour $|z|=1$, taken inthe anti-clockwise direciton , would be

1. $\frac{24\pi i}{13}$
2. $\frac{48\pi i}{13}$
3. $\frac{24}{13}$
4. $\frac{12}{13}$

Q. The value of integral $\underset{C}{\oint }\frac{si{n}^{2}z}{{(z-\frac{\pi }{2})}^3}dz$, where C is the |z| =1 , will be

1. $2\pi i$
2. $\pi i$
3. 0
4. $-\pi i$