If $\alpha $ and $\beta $ are different complex numbers with $|\beta | = 1$, then $\left| {\frac{{\beta - \alpha }}{{1 - \overline \alpha \beta }}} \right|$ is equal to

If the conjugate of $(x + iy)(1 - 2i)$ be $1 + i$, then

Let $z$ be a purely imaginary number such that ${\mathop{\rm Im}\nolimits} (z) < 0$. Then $arg\,(z)$ is equal to

If ${z_1} = 10 + 6i,{z_2} = 4 + 6i$ and $z$ is a complex number such that $amp\left( {\frac{{z - {z_1}}}{{z - {z_2}}}} \right) = \frac{\pi }{4},$ then the value of $|z - 7 - 9i|$ is equal to

If $z$ is a complex number such that $\frac{{z - 1}}{{z + 1}}$ is purely imaginary, then

Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$