Roots of Unity

If $\omega \left(\ne 1\right)$ is a cube root of unity and $\left(1+\omega \right){}^7=A+B\omega .$ Then, $\left(A, B\right)$ equals

$x^2+x+1$ is a factor of $a{x}^3+b{x}^2+cx+d=0$, then the real root of above equation is $(a, b, c, d\in R)$

If $\alpha =e^{\frac{2\pi i}{7}}$ and $f\left(x\right)=A_0+\sum _{k=1}^{20}{A}_k{x}^k,$ then find the value of
$f\left(x\right)+f\left(\alpha x\right)+\dots ..+f\left({\alpha }^{6}x\right)$ independent of $\alpha$

Given, $z=\cos \frac{2\pi }{2n+1}+i\sin \frac{2\pi }{2n+1}, \text{'}n\text{'}$ a positive integer, find the equation whose roots are, $\alpha =z+z^3+\dots ..+z^{2n-1}$ and $\beta =z^2+z^4+\dots ..+z^{2n}$.

Given $z_1+z_2+z_3=A, z_1+z_2\omega +z_3{\omega }^2=B, z_1+z_2{\omega }^2+z_3\omega =C,$ where $\omega$ is a cube root of unity,
(a) Express $z_1, z_2, z_3$ in terms of $A,B,C$.
(b) Prove that, ${\left|A\right|}^2+{\left|B\right|}^2+{\left|C\right|}^2=3\left({\left|z_1\right|}^2+{\left|z_2\right|}^2+{\left|z_3\right|}^2\right)$.
(c) Prove that $A^3+B^3+C^3-3ABC=27z_1{z}_2{z}_3$.

Let $z_k=\cos \left(\frac{2k\pi }{10}\right)+i\sin \left(\frac{2k\pi }{10}\right); k=1, 2,\dots , 9$.

Let $\mathrm{\omega }$ be a complex cube root of unity with $\omega \ne 1$ and $P=\left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij}={\omega }^{i+j}.$ Then $P^2\ne 0,$ when $n=$

Let $\omega =e^{\frac{2\pi i}{3}},$ and $a, b, c, x, y, z$ be non-zero complex numbers such that

$a+b+c=x$

$a+b\omega +c{\omega }^2=y$

$a+b{\omega }^2+c\omega =z.$

Then, the value of $\frac{|x{|}^2+|y{|}^2+|z{|}^2}{|a{|}^2+|b{|}^2+|c{|}^2}$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}=$

Which of the following is true?

If $\alpha$ is imaginary $n^{\mathrm{th}}, \left(n\ge 3\right)$ root of unity. Which of the following is/are true.

If $\alpha , \beta , \gamma$ are distinct roots of $x^3-3x^2+3x+7=0$ and $\omega$ is non-real cube root of unity, then the value of $\frac{\alpha -1}{\beta -1}+\frac{\beta -1}{\gamma -1}+\frac{\gamma -1}{\alpha -1}$ can be equal to

If $1,{\alpha }_1,{\alpha }_2,{\alpha }_3,\dots \dots ,{\alpha }_{n-1}$ be the $n^{\mathrm{th}}$ roots of unity, then which of the following are true

If $1,{\alpha }_1,{\alpha }_2,{\alpha }_3,\dots \dots ,{\alpha }_{n-1}$ be the $n^{\text{th }}$ roots of unity, then the value of $\sin \frac{\pi }{n}·\sin \frac{2\pi }{n}·\sin \frac{3\pi }{n}\dots \dots ..\sin \frac{\left(n-1\right)\pi }{n}$ equals

Let $\omega$ be the non real cube root of unity which satisfy the equation $h\left(x\right)=0$ where $h\left(x\right)=xf\left(x^3\right)+x^{2}g\left(x^3\right)$ If $h\left(x\right)$ is polynomial with real coefficient then which statement is incorrect.

If $p=a+b\omega +c{\omega }^2; q=b+c\omega +a{\omega }^2$ and $r=c+a\omega +b{\omega }^2$ where $a, b, c\ne 0$ and $\omega$ is the non-real complex cube root of unity, then

If $1, \omega \text{and} {\omega }^2$ are the cube roots of unity, then $\Delta =\left|\begin{array}{ccc}1& {\omega }^n& {\omega }^{2n}\\ {\omega }^n& {\omega }^{2n}& 1\\ {\omega }^{2n}& 1& {\omega }^n\end{array}\right|$ is equal to

If $x=a+b+c, y=a\alpha +b\beta +c$ and $z=a\beta +b\alpha +c$, where $\alpha$ and $\beta$ are imaginary cube roots of unity, then $xyz=$

Let $z_1$ and $z_2$ be two non real complex cube roots of unity and ${\left|z-z_1\right|}^2+{\left|z-z_2\right|}^2=\lambda$ be the equation of a circle with $z_1, z_2$ as ends of a diameter then the value of $\lambda$ is

If $\alpha$ is non real and $\alpha =\sqrt[5]{1}$, then the value of $2^{\left|1+\alpha +{\alpha }^2+{\alpha }^{-2}-{\alpha }^{-1}\right|}$ is equal to