If a complex number $z$ statisfies the equation $x + \sqrt 2 \,\,\left| {z + 1} \right|\,+ \,i\, = \,0,$ then $\left| z \right|$ is equal to

If $z$ is a complex number such that $| z | = 4$ and $arg \,(z) = \frac {5\pi }{6}$ , then $z$ is equal to

If ${z_1}$ and ${z_2}$ are two non-zero complex numbers such that $|{z_1} + {z_2}| = |{z_1}| + |{z_2}|,$then arg $({z_1}) – $arg $({z_2})$ is equal to

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

The number of solutions of the equation ${z^2} + \bar z = 0$ is