$1 . a$ is a non-zero real number, $x$ is a real number. Let proposition $M: x \in$ $\{-a, a\}$, let proposition $N: \sqrt{x^{2}}=a$ has a solution. Then $M$ is ( ) of $N$.
(A) sufficient but not necessary condition
(B) necessary but not sufficient condition
(C) sufficient and necessary condition
(D) neither sufficient nor necessary condition
$-、 1 . B$.

Let $a$ be a positive number. When $x=-a$, $x \in \{-a, a\}$, M is true. But $\sqrt{(-a)^{2}}=a \neq -a$, N is not true. Therefore, M is not a sufficient condition for N.

If N is true, $\sqrt{x^{2}}=a$, it is clear that $a$ should be a non-negative number. But $a$ is not 0, so $a$ is a positive number. Thus, $x=a \in \{-a, a\}$, so M is true. Therefore, M is a necessary condition for N.
In summary, M is a necessary but not a sufficient condition for N.