If $z $ is a complex number of unit modulus and  argument $\theta$, then ${\rm{arg}}\left( {\frac{{1 + z}}{{1 + (\bar z)}}} \right)$ equals.

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 $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that

The product of two complex numbers each of unit modulus is also a complex number, of

If the equation, $x^{2}+b x+45=0(b \in R)$ has conjugate complex roots and they satisfy $|z+1|=2 \sqrt{10},$ then

If ${z_1},{z_2} \in C$, then $amp\,\left( {\frac{{{{\rm{z}}_{\rm{1}}}}}{{{{{\rm{\bar z}}}_{\rm{2}}}}}} \right) = $