If $P$ is the point in the Argand diagram corresponding to the complex number $\sqrt{3}+i$ and if $OPQ$ is an isosceles right-angled triangle,right-angled at $O$,then $Q$ represents the complex number

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 $x^2+y^2+5x-3y+4=0$. Then which of the following statements is (are) $TRUE$?
$(A) \alpha=-1$ $(B) \alpha \beta=4$ $(C) \alpha \beta=-4$ $(D) \beta=4$

In the Argand plane,the distinct roots of $1+z+z^{3}+z^{4}=0$ ($z$ is a complex number) represent vertices of

For any real number $r$, let $A_r = \{e^{i \pi r n} : n \in \mathbb{N}\}$ be a set of complex numbers. Then,

If complex numbers ${z_1}, {z_2}, \text{and } {z_3}$ represent the vertices $A, B, \text{and } C$ respectively of an isosceles triangle $ABC$ of which $\angle C$ is a right angle,then the correct statement is:

$A$ man walks a distance of $3$ units from the origin towards the north-east $(N 45^{\circ} E)$ direction. From there,he walks a distance of $4$ units towards the north-west $(N 45^{\circ} W)$ direction to reach a point $P$. Then the position of $P$ in the Argand plane is