Let z be a complex number satisfying arg ((z-1)/(z+1))=(π)/(4). The maximum value of |z-2| is

Write $z=x+iy$. The argument of $(z-1)/(z+1)$ is the angle subtended at $z$ by the segment joining $-1$ and $1$. A constant angle $\pi/4$ means $z$ lies on an arc of a circle through $\pm1$ whose central angle is $\pi/2$, so the radius is $\sqrt2$ and the centre is $(0,1)$ (upper side, so the arc is the major arc with $y>0$). Thus $x^2+(y-1)^2=2$, $y>0$. The farthest point of the circle from $P=(2,0)$ lies on the ray from $P$ through the centre extended beyond it, at distance $|PC|+r=\sqrt5+\sqrt2$; this point is $C+\sqrt2\,(-2,1)/\sqrt5$, whose ordinate is positive, so it lies on the required arc. Hence the maximum is $\sqrt5+\sqrt2$. Option $\sqrt5+2$ wrongly uses radius $2$, and $\sqrt5+1$ uses radius $1$.