Let ${z_1}$ and ${z_2}$ be two complex numbers with $\alpha $ and $\beta $ as their principal arguments such that $\alpha + \beta > \pi ,$ then principal $arg\,({z_1}\,{z_2})$ is given by

If ${z_1}.{z_2}……..{z_n} = z,$ then $arg\,{z_1} + arg\,{z_2} + ….$+$arg\,{z_n}$ and $arg$$z$ differ by a

If $\frac{\pi }{2} < \alpha  < \frac{3}{2}\pi $ , then the modulus and argument of $(1 + cos\, 2\alpha ) + i\, sin\, 2\alpha $ is respectively

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 $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are $TRUE$?

$(A)$ $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$  $(B)$ $|z| \leq 2$ for all $z \in S$

$(C)$ $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$  $(D)$ The set $S$ has exactly four elements