Question

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

• A. 2
• B. 5
• C. 3
• D. 4

Answer 1

Answer: (C)

$ 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 $\Rightarrow \; $ is 5
$\alpha = 1 \Rightarrow \; \lambda \notin I$
$\alpha = 2 \Rightarrow \; \lambda \notin I$
$\alpha = 3 \Rightarrow \; \lambda \notin I \; \; \; \; \; \Rightarrow 3 $ integral values
$\alpha = 4 \Rightarrow \; \lambda \notin I$
$\alpha = 5 \Rightarrow \; \lambda \notin I$