The roots of $6 x^2-11 x+\alpha=0$ are rational numbers. ∴ Discriminant D must be perfect square number. $$ \begin{aligned} & D=(-11)^2-4 \cdot 6 \cdot \alpha \\ = & 121-24 \alpha \text { must be a perfect square } \\ \therefore & \alpha=3,4,5 \end{aligned} $$ $\therefore 3$ positive integral values are possible.