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

Read the following information carefully and answer the questions given below.

Let z₁, z₂ and z₃ be non-zero complex numbers satisfying $${z^2} = i\overline z $$, where $$i = \sqrt { - 1} $$

What is z₁ + z₂ + z₃ equal to?

Read the following information carefully and answer the questions given below.

Let z₁, z₂ and z₃ be non-zero complex numbers satisfying $${z^2} = i\overline z $$, where $$i = \sqrt { - 1} $$

Consider the following statements

1. $${z_1}{z_2}{z_3}$$ is purely imaginary.

2. $${z_1}{z_2} + {z_2}{z_3} + {z_3}{z_1}$$ is purely real.

Which of the above statements is/are correct?

Suppose, $$\omega$$ is a cube root of unity with $$\omega$$ $$\ne$$ 1. Suppose, P and Q are the points on the complex plane defined by $$\omega$$ and $$\omega$$². If O is the origin, then what is the angle between OP and OQ?