If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ and $\alpha, \beta$ are the complex cube roots of unity, then $x^3+y^3+z^3=$

If $z=\frac{3+2 i \cos \theta}{1-2 i \sin \theta}$ is a purely imaginary number, then

$$ \sin ^2 \theta+\cos ^2 3 \theta= $$

If $z=x+i y$ is a complex number such that $z \bar{z}^3+\bar{z} z^3=350$ and $x, y$ are integers, then $|z|=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then $(\alpha+\beta)^2+\left(\alpha^2+\beta^2\right)^2+\left(\alpha^3+\beta^3\right)^2+\ldots+\left(\alpha^{12}+\beta^{12}\right)^2=$