Let z_0 be a root of the quadratic equation,

Question:

Let $z_{0}$ be a root of the quadratic equation, $x^{2}+x+1=0$. If $z=3+6 i z_{0}^{81}-3 i z_{0}^{93}$, then arg $z$ is equal to:

  1. $\frac{\pi}{4}$

  2. $\frac{\pi}{3}$

  3. 0

  4. $\frac{\pi}{6}$


Correct Option: 1

Solution:

$z_{0}=\omega$ or $\omega^{2}$ (where $\omega$ is a non-real cube root of unity)

$z=3+6 i(\omega)^{81}-3 i(\omega)^{93}$

$z=3+3 i$

$\Rightarrow \arg z=\frac{\pi}{4}$

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