List of top Complex Analysis Questions on Residue Theorem asked in GATE MA

The value of the integral \[ \int_{C} \frac{z^{100}}{z^{101} + 1} \, dz \] where C is the circle of radius 2 centered at the origin taken in the anti-clockwise direction is

Let \( R = \{ z = x + iy \in \mathbb{C} : 0 < x < 1 \text{ and } - 11 \pi < y < 11 \pi \} \) and \( r \) be the positively oriented boundary of \( R \). Then the value of the integral \[ \frac{1}{2 \pi i} \int_r \frac{e^z}{e^z - 2} \, dz \] \text{is \(\underline{\hspace{1cm}}\) .}