If $${a_r} = {(\cos 2r\pi + i\sin 2r\pi )^{1/9}}$$,

then the value of $$\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|$$ is equal to

Let z₁ and z₂ be complex numbers such that z₁ $$ \ne $$ z₂ and |z₁| = |z₂|. If Re(z₁) > 0 and Im(z₂) < 0, then $${{{z_1} + {z_2}} \over {{z_1} - {z_2}}}$$ is

The expression $${{{{(1 + i)}^n}} \over {{{(1 - i)}^{n - 2}}}}$$ equals

Let z = x + iy, where x and y are real. The points (x, y) in the X-Y plane for which $${{{z + i} \over {z - i}}}$$ is purely imaginary, lie on