Question

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

• A. $2$
• B. $5$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$3$

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

given roots are rational

$\Rightarrow D$ must be the perfect square
$\Rightarrow 121-24\alpha ={\lambda }^2$

$\Rightarrow$ maximum value of $\alpha$ is $5$
$\alpha =1\Rightarrow \lambda \notin I$

$\alpha =2\Rightarrow \lambda \notin I$
$\alpha =3\Rightarrow \lambda \in I$

$\alpha =4\Rightarrow \lambda \in I$
$\alpha =5\Rightarrow \lambda \in I$ $\Rightarrow 3$ intergral value