The number of all possible positive integral values

$6 x^{2}-11 x+\alpha=0$

given roots are rational

$\Rightarrow$ D must be perfect square

$\Rightarrow 121-24 \alpha=\lambda^{2}$

$\Rightarrow$ maximum value of $\alpha$ is 5

$\alpha=1 \Rightarrow \lambda \notin \mathrm{I}$

$\alpha=2 \Rightarrow \lambda \notin \mathrm{I}$

$\alpha=3 \Rightarrow \lambda \in \mathrm{I}$                    $\Rightarrow 3$ integral values

$\alpha=4 \Rightarrow \lambda \in \mathrm{I}$

$\alpha=5 \Rightarrow \lambda \in \mathrm{I}$