Let $z$ satisfy $\left| z \right| = 1$ and $z = 1 – \vec z$.

Statement $1$ : $z$ is a real number

Statement $2$ : Principal argument of $z$ is $\frac{\pi }{3}$

Let $a \neq b$ be two non-zero real numbers.Then the number of elements in the set $X =\left\{ z \in C : \operatorname{Re}\left(a z^2+ bz \right)= a \text { and }\operatorname{Re}\left(b z^2+ az \right)= b \right\}$ is equal to

If ${z_1}$ and ${z_2}$ are any two complex numbers then $|{z_1} + {z_2}{|^2}$ $ + |{z_1} – {z_2}{|^2}$ is equal to

Let $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that

The set of all $\alpha  \in R$, for which $w = \frac{{1 + \left( {1 – 8\alpha } \right)z}}{{1 – z}}$ is a purely imaginary number, for all $z \in C$ satisfying $\left| z \right| = 1$ and ${\mathop{\rm Re}\nolimits} \,z \ne 1$,  is