The complex number $z$ satisfying $z+|z|$ $=1+7 i$, then the value of $|z|^2$ equals

If $$z_1, z_2$$ and $$z_3$$ are the vertices $$A, B$$ and $$C$$ respectively of an isosceles right angled triangle with right angled at $$C$$, then $$\left(z_1-z_3^{\prime}\right)\left(z_2-z_3\right)$$ equals to

If $$\alpha$$ is a non -real fifth root of unity, then the value of $$3^{\left|1+\alpha+\alpha^2+\alpha^{-2}-\alpha^{-1 \mid}\right|}$$, is

If $$\alpha, \beta$$ and $$\gamma$$ are the cube roots of $$P,(P<0)$$, then for any $$x, y$$ and $$z$$ which does not make denominator zero, the expression $$\frac{x \alpha+y \beta+z \gamma}{x \beta+y \gamma+z \alpha}$$ equals to