3. Given two sets of complex numbers
$$
\begin{array}{l}
M=\left\{z \mid z=\cos \alpha+\left(4-\cos ^{2} \alpha\right) \mathrm{i}, \alpha \in \mathbf{R}\right\}, \\
N=\{z \mid z=\cos \beta+(\lambda+\sin \beta) \mathrm{i}, \beta \in \mathbf{R}\} .
\end{array}
$$

When $M \cap N \neq \varnothing$, the range of the real number $\lambda$ is ( ).
(A) $[2,5]$
(B) $\left[\frac{11}{4}, 5\right]$
(C) $\left[\frac{11}{4},+\infty\right)$
(D) $(-\infty, 5]$
3. B.

From $M \cap N \neq \varnothing$, we know that the two sets must have common elements. Let $\cos \alpha+\left(4-\cos ^{2} \alpha\right) \mathrm{i}=\cos \beta+(\lambda+\sin \beta) \mathrm{i}$, then we have $\left\{\begin{array}{l}\cos \alpha=\cos \beta, \\ 4-\cos ^{2} \alpha=\lambda+\sin \beta .\end{array}\right.$
Eliminating $a$ gives
$$
\begin{array}{l}
\lambda=4-\cos ^{2} \beta-\sin \beta \\
=3+\sin ^{2} \beta-\sin \beta=\frac{11}{4}+\left(\sin \beta-\frac{1}{2}\right)^{2} .
\end{array}
$$

When $\sin \beta=\frac{1}{2}, \sin \beta=-1$, $\lambda$ respectively attains its minimum value $\frac{11}{4}$, and maximum value 5.