Complex Number

Consider the following statements in respect of the roots of the equation $x^3-8=0$ : The roots are non-collinear. The roots lie on a circle of unit radius. Which of the above statements is/are correct?

Suppose $\omega$ is a cube root of unity with $\omega \neq 1$. Suppose $P$ and Q are the points on the complex plane defined by $\omega$ and $\omega^2$. If $O$ is the origin, then what is the angle between OP and OQ ?

Which one of the following is correct in respect of the cube roots of unity?

If the point $z_1=1+i$ where $i=\sqrt{-1}$ is the reflection of $a$ point $z_2=x+i y$ in the line $i \bar{z}-i z=5$, then the point $z_2$ is

If $\alpha = \dfrac{-1 + \sqrt{-3}}{2}$, then what is the value of $(1 + \alpha^{19} - \alpha^{35})^{100} - (1 - 3\alpha^{25} + \alpha^{38})^{50}$?

Paragraph: Let $Z_1$ and $Z_2$ be any two complex numbers such that $Z_1^2+Z_2^2+Z_1 Z_2=0$ Question: What is the value of $\left|\frac{Z_1}{Z_2}\right|$ ?

What is the value of the sum $\sum_{\mathrm{n}=2}^{11}\left(\mathrm{i}^{\mathrm{n}}+\mathrm{i}^{\mathrm{n}+1}\right), \text { where } \mathrm{i}=\sqrt{-1} \text { ? }$

What is the value of $\sqrt{12+5 i}+\sqrt{12-5 i}$, where $i=\sqrt{-1}$ ?

The smallest positive integer $n$ for which $\left(\frac{1+i}{1-i}\right)^n=1$, is

Paragraph: Let $Z_1$ and $Z_2$ be any two complex numbers such that $Z_1^2+Z_2^2+Z_1 Z_2=0$ Question: What is the value of $\frac{1}{2}+\operatorname{Re}\left(\frac{\mathrm{Z}_1}{\mathrm{Z}_2}\right)$ ?

If $\omega$ is a non-real cube root of 1 , then what is the value of $\left|\frac{1-\omega}{\omega+\omega^2}\right| ?$

The number of roots of the equation $z^2=2 \bar{z}$ is