For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument with $-\pi<\arg ( z ) \leq \pi$. Then, which of the following statement (s) is (are) $FALSE$ ?

$(A)$ $\arg (-1- i )=\frac{\pi}{4}$, where $i =\sqrt{-1}$

$(B)$ The function $f: R \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-1+i t)$ for all $t \in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$

$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)\right.$ is an integer multiple of $2 \pi$.

$(D)$ For any three given distinct complex numbers, $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{\left( z – z _1\right)\left( z _2- z _3\right)}{\left( z – z _3\right)\left( z _2- z _1\right)}\right)=\pi$, lies on a straight line

If ${z_1}$ and ${z_2}$ are two complex numbers satisfying the equation $\left| \frac{z_1 +z_2}{z_1 – z_2} \right|=1$, then $\frac{{{z_1}}}{{{z_2}}}$ is a number which is

If $z = \frac{{ – 2}}{{1 + \sqrt 3 \,i}}$ then the value of $arg\,(z)$ is

The amplitude of the complex number $z = \sin \alpha + i(1 – \cos \alpha )$ is

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