If the conjugate of $(x + iy)(1 - 2i)$ be $1 + i$, then

If $z = 3 + 5i,\,\,{\rm{then }}\,{z^3} + \bar z + 198 = $

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?

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

If $arg\,z < 0$ then $arg\,( - z) - arg\,(z)$ is equal to

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

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