The minimum value of $|2z - 1| + |3z - 2|$ is

$Z$ is a complex number such that $|Z| \leq 2$ and $-\frac{\pi}{3} \leq \operatorname{amp} Z \leq \frac{\pi}{3}$. The area of the region formed by the locus of $Z$ is (in sq. units)

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

The locus of a point $z$ in the Argand plane that moves satisfying the equation $|z - (1 - i)| - |z - (2 + i)| = 3$ is:

If $\sin A+\sin B+\sin C=0$ and $\cos A+\cos B+\cos C=0$,then $\cos (A+B)+\cos (B+C)+\cos (C+A)$ is equal to

The values of $z$ for which $|z + i| = |z - i|$ are