Cube Roots of Unity

If $z^2+z+1=0,$ where $z$ is complex number, then the value of ${\left(z+\frac{1}{z}\right)}^2+{\left(z^2+\frac{1}{z^2}\right)}^2+{\left(z^3+\frac{1}{z^3}\right)}^2+\dots \dots +{\left(z^6+\frac{1}{z^6}\right)}^2$ is

If $a, b, c$ are the cube roots of $p<0,$ then for any $x, y, z:$ $\frac{x^2{a}^2+y^2{b}^2+z^2{c}^2}{x^2{b}^2+y^2{c}^2+z^2{a}^2}$ is equal to

$\left(1\right) 1$

$\left(2\right) a/c$

$\left(3\right)$ $b/a$

$\left(4\right)$ $c/b$

Correct options are-

If $\omega =\cos \frac{\pi }{\mathrm{n}}+i\sin \frac{\pi }{\mathrm{n}}\mathit{,}$ then value of $1+\omega +{\omega }^2+\dots +{\omega }^{n-1}$ is

If $\alpha , \beta$ are the roots of $x^2+px+q=0,$ and $w$ is an imaginary cube root of unity, then value of $\left(w\alpha +w^2\beta \right)\left(w^2\alpha +w\beta \right)$ is

If ${\left(1+x+x^2\right)}^n=a_0+a_{1}x+a_2{x}^2+\dots +a_{2n}{x}^{2n},$  then the value of $a_0+a_3+a_6+\dots$ is

If $x=a+b,y=a\omega +b{\omega }^2$ and $z=a{\omega }^2+b\omega ,$ then $x^3+y^3+z^3$ is equal to

If $x+iy={\left(1-i\sqrt{3}\right)}^{100},$ then find $\left(x,y\right)$

If $x=\omega -{\omega }^2-2,$ where $\omega \& {\omega }^2$ are cube roots of unity, then the value of $x^4+3x^3+2x^2-11x-6$ is

If $\omega$ is a cube root of unity, then $\sin \left\{\left({\omega }^{35}+{\omega }^{25}\right)\pi +\frac{\pi }{2}\right\}+\cos \left\{\left({\omega }^{10}+{\omega }^{23}\right)\pi -\frac{\pi }{4}\right\}$ is

If $\omega$ is imaginary cube root of unity, then $\sin \left\{\left({\omega }^{13}+{\omega }^2\right)\pi +\frac{\pi }{4}\right\}$ is equal to

The value of ${\left(1+\omega -{\omega }^2\right)}^7$ is, where $\omega$ is complex cube root of unity

If $\alpha , \beta$ are imaginary cube roots of unity, then ${\alpha }^4+{\beta }^{28}+\frac{1}{\alpha \beta }$ is equal to

If $x^2+x+1=0,$ then the value of  $\underset{x=1}{\overset{6}{\Sigma }}{\left(x^n+\frac{1}{x^n}\right)}^2$ is