If $$z$$ is a complex number, then the number of common roots of the equations $$z^{1985}+z^{100}+1=0$$ and $$z^3+2 z^2+2 z+1=0$$, is equal to

If $$z=x+i y, x y \neq 0$$, satisfies the equation $$z^2+i \bar{z}=0$$, then $$\left|z^2\right|$$ is equal to :

Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then $$(\mathrm{r}, \theta)$$ is equal to

If $$z=\frac{1}{2}-2 i$$ is such that $$|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$$ and $$\alpha, \beta \in \mathbb{R}$$, then $$\alpha+\beta$$ is equal to