Cube 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

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 $\alpha$ and $\beta$ are the distinct roots of equation $x^2-x+1=0$, then what is the value of $\left|\frac{{\alpha }^{100}+{\beta }^{100}}{{\alpha }^{100}-{\beta }^{100}}\right|$?

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

Let $\alpha , \beta , \gamma$ be the three  roots of the equation $x^3+bx+c=0$ if $\beta \gamma =1=-\alpha$ then $b^3+2c^3-3{\alpha }^3-6{\beta }^3-8{\gamma }^3$ is equal to 

Find the area of the triangle formed by roots of cubic equation ${\left(z+\alpha b\right)}^3={\alpha }^3\left(\alpha \ne 0\right)$.

If $\omega$ is imaginary cube root of unity, then value of $\sum _{r=0}^{54}\left(1+{\omega }^r+{\omega }^{2r}\right)$ equals to

If $1,\omega ,{\omega }^2$ are the cube roots of unity, $n\in \mathrm{\mathbb{N}}$ and $n>2$ then the least value of $n$ such that $1+\omega$ is a root of $x^n-x=0$ is

Let $a$ be a root of the equation $1+x^2+x^4 = 0$. Then the value of $a^{1011}+a^{2022}–a^{3033}$ is equal to:

The area of triangle whose vertices are $z, \omega z, z+\omega z$ is (where $\omega$ is complex cube root of unity)

If $\mathrm{\omega }$ is the cube root of unity, then $\left|\begin{array}{ccc}1& \mathrm{\omega }& {\mathrm{\omega }}^2\\ \mathrm{\omega }& {\mathrm{\omega }}^2& 1\\ {\mathrm{\omega }}^2& 1& \mathrm{\omega }\end{array}\right|=?$

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 $\omega =e^{-\frac{2\pi i}{3}},a,b,c,x,y,z$ are non-zero complex numbers such that $a+b+c=x,a+b\omega +c{\omega }^2=y$ and $a+b{\omega }^2+c\omega =z$, then $\frac{{\left|x\right|}^2+{\left|y\right|}^2+{\left|z\right|}^2}{{\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2}=$

Let $\omega$ be a cube root of $1$, which is different from $1$ Then the value of ${\left(1+{\omega }^{2021}\right)}^{2021}$ is

If $x=a+b+c, y=a\alpha +b\beta +c, z = a\beta +b\alpha +c$ where $\alpha , \beta$ are complex cube roots of unity and $a,b,c$ are real, then $xyz$ is equal to

If the cube roots of unity are $1, \omega , {\omega }^2$, then the roots of the equation ${\left(x+1\right)}^3+8=0$ are

The value of ${\left[\frac{-1+i\sqrt{3}}{2}\right]}^{-100}+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^{100}$ is

If $\omega$ is the cube root of unity, then the value of $\left(1-\omega \right)\left(1-{\omega }^2\right)\left(1-{\omega }^4\right)\left(1-{\omega }^8\right)$ is