The moduli of two complex numbers are less than unity, then the modulus of the sum of these complex numbers

Let $z$, $w \in C$ satisfy ${z^2} + \bar w = z$ and ${w^2} + \bar z = w$ then number of ordered pairs of  complex numbers $(z, w)$ is equal to

If $|{z_1}|\, = \,|{z_2}|$ and $arg\,\,\left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \pi $, then ${z_1} + {z_2}$ is equal to

If $\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi],$ is a real number, then an argument of $\sin \theta+\mathrm{i} \cos \theta$ is

Let $z_1, z_2 \in C$ such that $| z_1 + z_2 |= \sqrt 3$ and $|z_1| = |z_2| = 1,$ then the value of $|z_1 - z_2|$ is

If ${z_1}$ and ${z_2}$ are two complex numbers satisfying the equation $\left| \frac{z_1 +z_2}{z_1 - z_2} \right|=1$, then $\frac{{{z_1}}}{{{z_2}}}$ is a number which is