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GATE 2018 ME (ME1) – Question 14

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 1 mark · Multiple choice

For $z=x+iy$, $F(z)=iz+k\operatorname{Re}(z)+i\operatorname{Im}(z)$. Which k makes F satisfy the Cauchy–Riemann equations?

  1. 0
  2. 1
  3. −1
  4. y

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Correct answer: (B) 1

Explanation

Expand F: $iz=-y+ix$, so real part is $u=kx-y$ and imaginary part $v=x+y$. Cauchy–Riemann requires $u_x=v_y$, giving k = 1, while $u_y=-v_x$ already holds. Answer B.