If the set of all values of a, for which the equation has three distinct real roots, is the interval , then is equal to __________. [2025]

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Solution

Q.
If the set of all values of a, for which the equation $5 x^{3} - 15 x - a = 0$ has three distinct real roots, is the interval $(α, β)$ , then $β - 2 α$ is equal to __________. [2025]

Ans.
(30)

Given, $5 x^{3} - 15 x - a = 0 \Rightarrow a = 5 x^{3} - 15 x$

Let $P (x) = 5 x^{3} - 15 x$

Differentiating w.r.t. x, we get

$P' (x) = 15 x^{2} - 15 = 15$

$(x - 1) (x + 1)$

$∴ x \in (- 10, 10)$

$∴ β - 2 α = 10 - 2 (- 10) = 10 + 20 = 30$ .

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