1. If $M=\left\{z\left|(z-1)^{2}=\right| z-\left.1\right|^{2}, z \in \mathbf{C}\right\}$, then
A. $M=\{$ real numbers $\}$
B. $M=\{$ imaginary numbers $\}$
C. $\{$ real numbers $\} \varsubsetneqq M \varsubsetneqq\{$ complex numbers $\}$
D. $M=\{$ complex numbers $\}$
1. A Let $z=x+yi \quad(x-1)^{2}-y^{2}+2yi(x-1)=(x-1)^{2}+y^{2} \quad$ so $y=0$.