### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Fundamentals of Complex analysis** under *Complex analysis* in GATE PH. 2. **Stage 2: Mathematical Derivation:** * $\oint_{|z|=2} \frac{dz}{z-1} = 2\pi i \text{Res}(f, 1) = 2\pi i (1) = 2\pi i$. Option A is correct. 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Evaluating the integral as zero assuming Cauchy's Integral Theorem without checking if poles lie inside the contour. *Reference: George B. Arfken, Hans J. Weber, Mathematical Methods for Physicists (Academic Press).*