Question

If $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1$ and $z_{1}, z_{2}, z_{3}$ are represented by the vertices of an equilateral triangle then which of the following is not true?

• A. $z_{1}+z_{2}+z_{3}=0$
• B. $z_{1} z_{2} z_{3}=1$
• C. $z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=0$
• D. None of these

Answer 1

Answer: (B)

All the three vertices lies on circle $|z|=1$.
Take $z_{1}=z_{1}, z_{2}=z_{1} \omega, z_{3}=z_{1} \omega^{2}$
So $z_{1}+z_{2}+z_{3}=z_{1}\left(1+\omega+\omega^{2}\right)=0$
$z_{1} z_{2} z_{3}=z_{1}^{3} $
$z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=z_{1}^{2}\left(\omega+\omega^{3}+\omega^{2}\right)=0$