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SSC Pure Mathematics QUESTION #4128
Question 1
For $\oint_{|z|=2} \frac{5z - 2}{z(z-1)} \, dz$, what is the value?
  • $0$
  • $6\pi i$
  • $10\pi i$✔️
  • $4\pi i$
Correct Answer Explanation
Residue at $z=0$: $\lim_{z \to 0} z \cdot \frac{5z-2}{z(z-1)} = \frac{-2}{-1} = 2$. Residue at $z=1$: $\lim_{z \to 1} (z-1) \cdot \frac{5z-2}{z(z-1)} = \frac{3}{1} = 3$. Sum of residues $= 5$. Integral $= 2\pi i \times 5 = 10\pi i$.