The number of non-zero integral solution of the equation $$|1 - 2i{|^x} = {5^x}$$ is

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

What is $${i^{1000}} + {i^{1001}} + {i^{1002}} + {i^{1003}}$$ equal to (where, $$i = \sqrt { - 1} $$)?

The modulus-amplitude form of $$\sqrt 3 + i$$, where $$i = \sqrt { - 1} $$ is