Roots of Unity

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 ­–

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 

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 $1,z_1,z_2,\dots \dots ,z_{n-1}$ be the $n^{\text{th }}$ roots of unity and '$\omega$' be a non-real complex cube root of unity, then sum of all possible values of $\prod _{r=1}^{n-1}\left(\omega -z_r\right)$ will be equal to

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 $\omega$ 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 $\frac{{\left|x\right|}^2+{\left|y\right|}^2+{\left|z\right|}^2}{{\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2}=$

The common roots of the equations $Z^3+2z^2+2z+1=0, z^{2014}+z^{2015}+1=0$ are

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

The roots of the equation $x^3=1$ are

If $1, \omega , {\omega }^2$ are three cube roots of unity, then ${\left(1-\omega +{\omega }^2\right)}^3=$