What is the value of $${\left( {{{ - 1 + i\sqrt 3 } \over 2}} \right)^{3n}} + {\left( {{{ - 1 - i\sqrt 3 } \over 2}} \right)^{3n}}$$, where $$i = \sqrt { - 1} $$?

Which one of the following is correct in respect of the cube roots of unity?

What is the principal argument of ($$-$$1 $$-$$i), where $$i = \sqrt { - 1} $$?

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