What is $${\omega ^{100}} + {\omega ^{200}} + {\omega ^{300}}$$ equal to, where $$\omega$$ is the cube root of unity?

If $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {z + 1}}} \right) = 0$$, where z = x + iy is a complex number, then which one of the following is correct?

If $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^{107}} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^{107}}$$, then what is the imaginary part of z equal to?

What is the number of distinct solutions of the equation $${z^2} + |z|\, = \,0$$ (where, z is the complex number)?