Find the modulus and argument of the complex number $\frac{1+2 i}{1-3 i}$

Let $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that

If $z$ is a purely real number such that ${\mathop{\rm Re}\nolimits} (z) < 0$, then $arg(z)$ is equal to

If $z_1 , z_2$ and $z_3, z_4$ are $2$ pairs of complex conjugate numbers, then $\arg \left( {\frac{{{z_1}}}{{{z_4}}}} \right) + \arg \left( {\frac{{{z_2}}}{{{z_3}}}} \right)$ equals 

Let $S=\left\{z \in C : z^{2}+\bar{z}=0\right\}$. Then $\sum \limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to$……$