Let $z$, $w \in C$ satisfy ${z^2} + \bar w = z$ and ${w^2} + \bar z = w$ then number of ordered pairs of  complex numbers $(z, w)$ is equal to

Amplitude of $\left( {\frac{{1 – i}}{{1 + i}}} \right)$ is

If ${z_1} = 1 + 2i$ and ${z_2} = 3 + 5i$, and then $\operatorname{Re} \left( {\frac{{{{\bar z}_2}{z_1}}}{{{z_2}}}} \right)$ is equal to

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

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?

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