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:

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Quadratic Equations and Inequalities

The number of all possible positive integral values of

α

for which the roots of the quadratic equation,

6 x^{2} - 11 x + α = 0

are rational numbers is:

Answer: (C)

Rational roots of

6 x^{2} - 11 x + α = 0

require the discriminant

D = 121 - 24 α

to be a non-negative perfect square. Since

D \geq 0

forces

α \leq 5

, test the positive integers:

α = 3

gives

D = 49 = 7^{2}

α = 4

gives

D = 25 = 5^{2}

, and

α = 5

gives

D = 1 = 1^{2}

, while

α = 1, 2

give

97

and

73

, which are not perfect squares. So there are

3

admissible values.

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option