$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$$ is equal to

For positive integers $${n_1},\,{n_2}$$ the value of the expression $${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$$
where $$i = \sqrt { - 1} $$ is real number if and only if

Let $$z$$ and $$\omega $$ be two non zero complex numbers such that
$$\left| z \right| = \left| \omega \right|$$ and $${\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,$$ then $$z$$ equals

Let $$z$$ and $$\omega $$ be two complex numbers such that
$$\left| z \right| \le 1,$$ $$\left| \omega \right| \le 1$$ and $$\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2$$ then $$z$$ equals