2. If $M=\left\{(x, y)|| \tan \pi y \mid+\sin ^{2} \pi x=0\right\}, N=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2\right\}$, then $|M \cap N|=$ ( )
A. 4
B. 5
C. 8
D. 9
2. D From $\tan \pi y=0$ we get $y=k, k \in \mathbf{Z}$; from $\sin \pi x=0$ we get $x=k', k' \in \mathbf{Z}$; also, $x^{2}+y^{2} \leqslant 2$, so $k=0$, $-1,1 ; k'=0,-1,1$. Therefore, $M \cap N$ contains a total of 9 points.