Let $z_1$ and $z_2$ be two distinct complex numbers let $z=(1-t) z_1+t z_2$ for some real number t with $0 < t < 1$.

If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$, then :

If $${\omega}$$ is an imaginary cube root of unity, then $${(1\, + \omega \, - {\omega ^2})^7}$$ equals

If $$\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$$ , then

The value of the sum $$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$ , where i = $$\sqrt { - 1} $$, equals