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:

8 \sqrt{3}

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4 \sqrt{3}

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10 \sqrt{5}

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8 \sqrt{5}

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Solution

The correct option is D $8 \sqrt{5}$

$S O R = \frac{3}{m^{2} + 1}$

$\Rightarrow {(S . O . R)}_{m a x} = 3$ ........................when $m = 0$

$α + β = 3; α β = 1$

$∣ ∣ α^{3} - β^{3} ∣ ∣ = | α - β | ∣ ∣ (α^{2} + β^{2} + α β) ∣ ∣ = ∣ ∣ ∣ \sqrt{{(α - β)}^{2} - α β} ({(α - β)}^{2} - α β) ∣ ∣ ∣ = ∣ ∣ \sqrt{9 - 4} (9 - 1) ∣ ∣ = \sqrt{5} \times 8$

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