If the equation of the median through vertex $A(3, k)$ of $\triangle A B C$ with vertices $\mathrm{B}(2,1)$ and $\mathrm{C}(-4,5)$ is $x+4 \mathrm{y}=\mathrm{p}$, then $\mathrm{k}=$ where p and k are constants

The median from vertex $A(3, k)$ passes through the midpoint $D$ of side $BC$, with $B(2, 1)$ and $C(-4, 5)$.

The midpoint $D$ is given by $\left(\frac{2 + (-4)}{2}, \frac{1 + 5}{2}\right) = (-1, 3)$.

Since $D$ lies on the median $x + 4y = p$, substitution yields $-1 + 4(3) = 11$, so $p = 11$.

Vertex $A(3, k)$ must also satisfy $x + 4y = 11$, giving $3 + 4k = 11$, which simplifies to $4k = 8$ and thus $k = 2$.

Final answer: $\boxed{2}$