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]

Question

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]

Smart Achievers · Accepted Answer

(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

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

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

$∴ x \in (- 10, 10)$

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

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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Solution

Q. If the set of all values of a, for which the equation 5x3–15x–a=0 has three distinct real roots, is the interval (α, β), then β–2α is equal to __________. [2025]

Ans. (30) Given, 5x3–15x–a=0 ⇒ a=5x3–15x Let P(x)=5x3–15x Differentiating w.r.t. x, we get P'(x)=15x2–15=15 (x–1)(x+1) ∴ x∈(–10,10) ∴ β–2α=10–2(–10)=10+20=30.

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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

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

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

$∴ x \in (- 10, 10)$

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