For $n \in \mathbb{N}$, a complex number $\omega$ is called an $n$-th root of unity if $\omega^n=1$. Then

• A

  the product of all the distinct $n$-th roots of unity is $(-1)^n$.
• B

  $\sum_{j=0}^4\left|z_1+\omega^{2 j} z_2\right|^2=5\left(\left|z_1\right|^2+\left|z_2\right|^2\right) \quad$ for any $z_1, z_2 \in \mathbf{C}$.
• Correct answer
  C

  if $n \geq 5$ and $\omega$ is an $n$-th root of unity different from 1 then
  $\sum_{j=0}^{n-1}\left|z_1+\omega^j z_2\right|^2=n\left(\left|z_1\right|^2+\left|z_2\right|^2\right) \quad \text { for any } z_1, z_2 \in \mathbb{C}$
• Correct answer
  D

  if $n \geq 5$ then the complex conjugate of an $n$-th roots of unity is an $n$-th root of unity.

Hence, the correct answer is options: 'C', 'D'.