Let S be the set of complex numbers z satisfying arg ((z+1)/(z-1))=(π)/(2).

Put $z=x+iy$. Then $\frac{z+1}{z-1}=\frac{(z+1)(\bar z-1)}{|z-1|^2}=\frac{|z|^2-1-2iy}{|z-1|^2}$. This is a positive multiple of $i$ exactly when $|z|=1$ and $y<0$, so $S$ is the open lower unit semicircle (endpoints $\pm1$ excluded). (A) For $z=-i$: $\frac{1-i}{-1-i}=i$, so $-i\in S$: correct. (B) For $z=e^{i\theta}$ with $\sin\theta\in(-1,0)$, $|z-2i|^2=5-4\sin\theta\in(5,9)$; the value $\sqrt5$ would need $\sin\theta=0$, i.e. an excluded endpoint, so it is not attained: incorrect. (C) The supremum $3$ occurs at $\sin\theta=-1$, i.e. $z=-i\in S$, so the maximum is $3$: correct. (D) For $|z|=1$, $|z-1|^2+|z+1|^2=2|z|^2+2=4$: correct. Answer: A, C and D.