$Amp\left(\frac{Z+1}{Z-2i}\right)=\frac{\pi }{2}$  then locus of Z = x + iy is

$$\begin{gathered} Amp\left[\frac{(x+1)+iy}{x+i(y-2)}\right]=\frac{\pi }{2} \\ \Rightarrow Amp \left(\frac{x\left(x+1\right)+y\left(y-2\right)+i\left(xy-\left(x+1\right)\left(y-2\right)\right)}{x^2+{\left(y-2\right)}^2}\right)=\frac{\mathrm{\pi }}{2} \end{gathered}$$

 $\Rightarrow x\left(x+1\right)+y\left(y-2\right)=0,xy-\left(x+1\right)\left(y-2\right)>0$ 

                                          i.e   2x – y + 2 > 0

 which is the required locus.