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GATE 2014 EE (EE1) – Question 15

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy integral formula, Taylor series, Laurent series, Residue theorem · 1 mark · Multiple choice

Let $S=\{z:|z|=1\}$ in the complex plane. Where does $f(z)=zz^*$, with conjugate z*, map S?

  1. Unit circle
  2. Horizontal-axis segment from (0,0) to (1,0)
  3. The point (1,0)
  4. Entire horizontal axis

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Show answer and explanation

Correct answer: (C) The point (1,0)

Explanation

Complex conjugation gives zz*=|z|². Every point of the unit circle has |z|=1, so all of them map to the real number1. The image is the **single point(1,0),C**, not a circle or interval.