Evaluate C z100/(z101+1)\,dz, where C is z =2 counterclockwise.

GATE 2022 MA – Question 14

Complex Analysis · Complex Integration and Residues · 1 mark · Multiple choice

Evaluate $\oint_C z^{100}/(z^{101}+1)\,dz$, where C is |z|=2 counterclockwise.

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Correct answer: (D) 2πi

All 101 simple roots of z¹⁰¹+1 lie on the unit circle, within C. Each residue is z¹⁰⁰/(101z¹⁰⁰)=1/101, so the residue sum is 1. Integral=**2πi, D**.

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