If m is chosen in the quadratic equation m 2 + 1 x 2 - 3 x + m 2 + 1 2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

Step-by-step solution

Let, the roots of m 2 + 1 x 2 - 3 x + m 2 + 1 2 = 0 are α and β We know that the sum and product of roots of a quadratic equation a x 2 + b x + c = 0 ,   a ≠ 0 are respectively - b a and c a . ∴   α + β = 3 m 2 + 1 and α β = m 2 + 1 2 m 2 + 1 = m 2 + 1 ∵   α + β is maximum, ∴   m = 0 ∴ the equation is x 2 - 3 x + 1 = 0 And α + β = 3 ,   α β = 1 Now, α 3 - β 3 = α - β α 2 + &#94