The complex numbers $sin\ x + i\ cos\ 2x$ and $cos\ x\ -\ i\ sin\ 2x$ are conjugate to each other, for

For any complex number $w = c + id$, let $\arg ( w ) \in(-\pi, \pi]$, where $i =\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+$ iy satisfying arg $\left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, the ordered pair $( x , y )$ lies on the circle

$x^2+y^2+5 x-3 y+4=0 .$

Then which of the following statements is (are) TRUE?

$(A)$ $\alpha=-1$  $(B)$ $\alpha \beta=4$   $(C)$ $\alpha \beta=-4$   $(D)$ $\beta=4$

Let ${z_1}$ be a complex number with $|{z_1}| = 1$ and ${z_2}$be any complex number, then $\left| {\frac{{{z_1} - {z_2}}}{{1 - {z_1}{{\bar z}_2}}}} \right| = $

If $z$ is a complex number such that $\frac{{z - 1}}{{z + 1}}$ is purely imaginary, then

The inequality $|z - 4|\, < \,|\,z - 2|$represents the region given by

If $z$ is a complex number such that $|z - \bar{z}| = 2$ and $|z + \bar{z}| = 4 $, then which of the following is always incorrect -