The number of all possible positive integral values of $\alpha$  for which the roots of the quadratic equation, $6x^2-11x+\alpha =0$   are rational numbers is

 $6x^2-11x+\alpha =0$
Given roots are rational
$\Rightarrow D$ must be perfect square
 $\Rightarrow 121-24\alpha ={\lambda }^2$
$\Rightarrow$ maximum value of $\alpha$  is 5
 $$\begin{gathered} \alpha =1\Rightarrow \lambda \notin I \\ \alpha =2\Rightarrow \lambda \notin I \\ \alpha =3\Rightarrow \lambda \in I \\ \alpha =4\Rightarrow \lambda \in I \\ \alpha =5\Rightarrow \lambda \in I \end{gathered}$$
  Number of values = 3