2. Let condition $p$ :

real numbers $m, n$ satisfy $\left\{\begin{array}{l}2<m+n<4, \\ 0<m n<3 ;\end{array}\right.$
condition $q$ :
real numbers $m, n$ satisfy $\left\{\begin{array}{l}0<m<1, \\ 2<n<3 .\end{array}\right.$ Then the correct option is ( ).
(A) $p$ is a sufficient but not necessary condition for $q$
(B) $p$ is a necessary but not sufficient condition for $q$
(C) $p$ is a sufficient and necessary condition for $q$
(D) $p$ is neither a sufficient condition nor a necessary condition for $q$
2. B.

From condition $q$, we know
$$
2<m+n<4,0<m n<3 \text {. }
$$

Therefore, $p$ is a necessary condition for $q$.
Notice that, when $m=1, n=2$, condition $p$ is satisfied, but condition $q$ is not.
Thus, $p$ is not a sufficient condition for $q$.
In conclusion, $p$ is a necessary but not sufficient condition for $q$.