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GATE 2018 ME (ME2) – Question 36

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

For a complex variable z and counterclockwise integration around the unit circle C centred at the origin, $\oint_C\frac{1}{5z-4}\,dz=A\pi i$. Find A.

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

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Correct answer: (A) 2/5

Explanation

The pole $z=4/5$ lies inside C and has residue 1/5. Counterclockwise integration gives $2\pi i(1/5)=2\pi i/5$. Compare with $A\pi i$: **A = 2/5, option A**.