Roots of Unity

Let  $\omega$ be a complex cube root of unity with  $\omega \ne 1\mathrm{.}$  A fair die is thrown three times.  $r_1,r_2\mathrm{and}{r}_3$  are the numbers obtained on the die, then the probability that  ${\omega }^{r_1}+{\omega }^{r_2}+{\omega }^{r_3}=0$ is

Let $z_1$  and  $z_2$ be nth roots of unity which subtend a right angle at the origin, then n must be of the form -

If $\omega$ is an imaginary cube root of unity and  ${(1+\omega )}^7=A+B\omega$, then real numbers $A$ and $B$ are respectively ­–

If $\omega$ is a cube root of unity, then the value of polynomial $\left|\begin{array}{ccc}x+1& \omega & {\omega }^2\\ \omega & x+{\omega }^2& 1\\ {\omega }^2& 1& x+\omega \end{array}\right|$ is

If $x^2+x+1=0$, then $-\left\{{\left(x-\frac{1}{x}\right)}^2+{\left(x^2-\frac{1}{x^2}\right)}^2+{\left(x^3-\frac{1}{x^3}\right)}^2\right\}$ is ___________.

If $\omega$ is a cube root of unity, then $1+{\omega }^2=$

If $z^2+z+1=0$, where $z$ is a complex number, find the value of  ${\left(z+\frac{1}{z}\right)}^2+{\left(z^2+\frac{1}{z^2}\right)}^2+{\left(z^3+\frac{1}{z^3}\right)}^2+{\left(z^4+\frac{1}{z^4}\right)}^2+{\left(z^5+\frac{1}{z^5}\right)}^2+{\left(z^6+\frac{1}{z^6}\right)}^2$.

Find the number of ${15}^{\mathrm{th}}$ roots of unity, which are also ${25}^{\mathrm{th}}$ roots of unity.

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then find the value of $k$ if  $(a+2b{)}^2+{\left(a{\omega }^2+2b\omega \right)}^2+{\left(a\omega +2b{\omega }^2\right)}^2=kab$.

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then find the value of $(2-\omega )\left(2-{\omega }^2\right)\left(2-{\omega }^{10}\right)\left(2-{\omega }^{11}\right)$ is 

If ${\left(1+\omega \right)}^7=A+B\omega$ then

On any given arc of positive length on the unit circle $\left|z\right|=1$ in the complex plane.

The value of $\sum _{\mathrm{k}=1}^{99}\left({\mathrm{i}}^{\mathrm{k}!}+{\mathrm{\omega }}^{\mathrm{k}!}\right)$ is (where, $\mathrm{i}=\sqrt{-1}$ and $\mathrm{\omega }$ is non-real cube root of unity)

If $\alpha , \beta \in C$ are the distinct roots of the equation $x^2-x+1=0,$ then ${\alpha }^{101}+{\beta }^{107}$ is equal to

Evaluate :  ${\left(-\frac{1}{2}+\frac{\sqrt{3}i}{2}\right)}^{1000}$.

Evaluate ${\left(1+\omega -{\omega }^2\right)}^7$, where $\omega$ is an imaginary cube root of unity.

Common roots between equations $x^5-x^3+x^2-1=0$ and  $x^4=1$ are____

If $\omega$ is a complex cube root of unity, then find
the value of ${\left(\frac{a+b\omega +c{\omega }^2}{c+a\omega +b{\omega }^2}+\frac{a+b\omega +c{\omega }^2}{b+c\omega +a{\omega }^2}\right)}^2.$

Find the sum of common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ :