The real value of $\theta$ for which the expression $\frac{{1 + i\,\cos \theta }}{{1 - 2i\cos \theta }}$ is a real number is $\left( {n \in I} \right)$ 

If $|z|\, = 1,(z \ne - 1)$and $z = x + iy,$then $\left( {\frac{{z - 1}}{{z + 1}}} \right)$ is

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 $z=\frac{1}{2}-2 i$, is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in R \quad$, then $\alpha+\beta$ is equal to

The conjugate of a complex number is $\frac{1}{{i - 1}}$ then that complex number is

If $|z|\, = 4$ and $arg\,\,z = \frac{{5\pi }}{6},$then $z =$