If f x -0 is a cubic equation with positive and distinct roots - - Y such that - is H-M of the roots of f - x -0- then - - and Y are inA- A-PB- G-PC- H-PD- none of these

The correct option is B G.Pf(x)=(x α)(x β)(x γ) ⇒ f'(x)=3x^2 2x(α +β +γ )+αβ +βγ +γα ⇒β =2 α_1 β_1/α_1+β_1 (where α_1, β_1 are the roots of f'(x) =0) ⇒β^2=γα ...

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Standard XII

Mathematics

Geometric Progression

If f x =0 is ...

Question

If f(x) = 0 is a cubic equation with positive and distinct roots $α, β, γ$ such that $β$ is H.M of the roots of $f^{'} (x) = 0$ , then $α, β$ and $γ$ are in

A.P

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G.P

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H.P

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none of these

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Solution

The correct option is B G.P
$f (x) = (x - α) (x - β) (x - γ) \Rightarrow f^{'} (x) = 3 x^{2} - 2 x (α + β + γ) + α β + β γ + γ α$
$\Rightarrow β = \frac{2 α_{1} β_{1}}{α_{1} + β_{1}}$ (where $α_{1}, β_{1}$ are the roots of $f^{'} (x) = 0)$
$\Rightarrow β^{2} = γ α$

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If f(x) = 0 is a cubic equation with positive and distinct roots α,β,γ such that β is H.M of the roots of f′(x)=0, then α,β and γ are in

The correct option is B G.Pf(x)=(x−α)(x−β)(x−γ)⇒f′(x)=3x2−2x(α+β+γ)+αβ+βγ+γα ⇒β=2α1β1α1+β1 (where α1,β1 are the roots of f′(x)=0) ⇒β2=γα

If f(x) = 0 is a cubic equation with positive and distinct roots $α, β, γ$ such that $β$ is H.M of the roots of $f^{'} (x) = 0$ , then $α, β$ and $γ$ are in

The correct option is B G.P
$f (x) = (x - α) (x - β) (x - γ) \Rightarrow f^{'} (x) = 3 x^{2} - 2 x (α + β + γ) + α β + β γ + γ α$
$\Rightarrow β = \frac{2 α_{1} β_{1}}{α_{1} + β_{1}}$ (where $α_{1}, β_{1}$ are the roots of $f^{'} (x) = 0)$
$\Rightarrow β^{2} = γ α$