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

Let $z_1,z_2,\dots ,z_7$ be the vertices of a regular heptagon that is inscribed in the unit circle with centre at the origin in the complex plane. Let $w=\sum _{1\le i<j\le 7}{z}_i{z}_j$, then $\left|w\right|$ is equal to

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 $z=\frac{\sqrt{3}}{2}+\frac{i}{2} \left(i=\sqrt{-1}\right),$ then ${\left(1+iz+z^5+i{z}^8\right)}^9$ is equal to:

If $1, \omega , {\omega }^2,\dots . {\omega }^{n-1}$ are $n, n^{th}$ roots of unity, then the then the value $\left(9-\omega \right)\left(9-{\omega }^2\right)\left(9-{\omega }^3\right)\dots . \left(9-{\omega }^{n-1}\right)$ is

If $\omega$ is an imaginary cube root of unity than $\sqrt{-1-\sqrt{-1-\sqrt{-1-\dots \mathrm{t}\mathrm{o}\mathrm{ }\mathrm{\infty }}}}$ can be equal to

If $i=\sqrt{-1}$, then $4+5{\left(-\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)}^{334}+3 {\left(-\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)}^{365}$ is equal to

Let, $z_k (k=\mathrm{0,\; 1}, 2, \dots .., 6)$ be the roots of the equation ${\left(z+1\right)}^7+z^7=0$, then $\sum _{k=0}^{6}Re\left(z_k\right)$ is equal to

If $z+\frac{1}{z}+1=0,$ then $z^{2003}+\frac{1}{z^{2003}}$ is equal to 

Let $\mathrm{w}\ne 1$ and ${\mathrm{w}}^{13}=1,$ then find the Quadratic equation whose roots are $\mathrm{w}+{\mathrm{w}}^3+{\mathrm{w}}^4+{\mathrm{w}}^{-4}+{\mathrm{w}}^{-3}+{\mathrm{w}}^{-1}$ and ${\mathrm{w}}^2+{\mathrm{w}}^5+{\mathrm{w}}^6+{\mathrm{w}}^{-6}+{\mathrm{w}}^{-5}+{\mathrm{w}}^{-2}$. 

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

Let $\omega$ be a complex number such that $2\omega +1=z$ where $z=\sqrt{-3}$ . If

$\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\omega }^2-1& {\omega }^2\\ 1& {\omega }^2& {\omega }^7\end{array}\right|=3k,$

Then k can be equal to:

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

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

The value of $\frac{a+b\omega +c{\omega }^2}{b+c\omega +a{\omega }^2}+\frac{a+b\omega +c{\omega }^2}{c+a\omega +b{\omega }^2}$ will be: ($\omega$ is non real cube root of unity and expression is well defined)