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GATE 2021 EE – Question 12

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

Let $p(z)=z^3+(1+j)z^2+(2+j)z+3$, where $z$ is a complex number. Which one of the following is true?

  1. conjugate$\{p(z)\}=p(\text{conjugate}\{z\})$ for all $z$
  2. The sum of the roots of $p(z)=0$ is a real number
  3. The complex roots of the equation $p(z)=0$ come in conjugate pairs
  4. All the roots cannot be real

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Correct answer: (D) All the roots cannot be real

Explanation

The coefficients of $p(z)$ are not all real, so conjugate symmetry (A) and conjugate-pair roots (C) need not hold. The sum of the roots is $-(1+j)$, which is not real (B is false). If all three roots were real their sum would be real, so all the roots cannot be real.