1. If $M=\left\{(x, y)|| \operatorname{tg} \pi y \mid+\sin ^{2} \pi x=0\right\}$,
$$
N=\left\{(x, y)\left|x^{2}+y^{2}\right| \leqslant 2\right\},
$$

then the number of elements in $M \cap N$ is
(A) 4 ;
(B) 5 ;
(C) 8 ;
(D) 9 .
1. (D)

From $\operatorname{tg} \pi y=0$ we get
$$
y=k(k \in \mathbb{Z}), \sin \pi x=0
$$

we get
$$
\begin{array}{ll} 
& x=k^{\prime}\left(k^{\prime} \in \mathbb{Z}\right) . \\
\text { Also } & x^{2}+y^{2} \leqslant 2 . \\
\therefore \quad & k=-1,0,1 ; \\
& k^{\prime}=-1,0,1 .
\end{array}
$$

As shown in the figure, there are 9 points in total, so the answer is (D).