Question:
The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $6 x^{2}-11 x+\alpha=0$ are rational numbers is :
Correct Option: , 3
Solution:
$6 x^{2}-11 x+\alpha=0$
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
$\alpha=4 \Rightarrow \lambda \in \mathrm{I}$
$\alpha=5 \Rightarrow \lambda \in \mathrm{I}$