If - are the roots of x2-px-1-0- - - the roots of x2 - qx - 1 - 0- then the value of - is

Α+β= p , αβ=1 Also γ ^2 + 1= qγ nbsp; and δ^2 +1= qδ (α γ)( β γ)(α+ δ)( β +δ) = (αβ (α+β)γ+γ^2)(αβ+(α+β)δ+δ^2) =(γ^2+pγ+1)(δ^2 pδ+1) =(pγ qγ)( pδ qδ) ...

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Byju's Answer

Standard XI

Mathematics

Relations Between Roots and Coefficients

If α,β are th...

Question

If $α, β$ are the roots of $x^{2} + p x + 1 = 0,$ $δ, γ$ the roots of $x^{2} + q x + 1 = 0,$ then the value of $(α - γ) (β - γ) (α + δ (β + δ))$ is

q²

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p^{2}

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q² - p²

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p² - q²

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Solution

The correct option is C
q² - p²

$α + β = - p$ , $α β = 1$
Also $γ^{2} + 1 = - q γ$ and $δ^{2} + 1 = - q δ$

$(α - γ) (β - γ) (α + δ) (β + δ) = (α β - (α + β) γ + γ^{2}) (α β + (α + β) δ + δ^{2}) = (γ^{2} + p γ + 1) (δ^{2} - p δ + 1) = (p γ - q γ) (- p δ - q δ) = (q^{2} - p^{2}) γ δ = (q^{2} - p^{2})$

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Similar questions

Q. If α, β are the roots of the equation

x^{2} + p x + 1 = 0; γ, δ

the roots of the equation

x^{2} + q x + 1 = 0, t h e n (α - γ) (α + δ) (β - γ) (β + δ) =

(a)

q^{2} - p^{2}

(b)

p^{2} - q^{2}

(c)

p^{2} + q^{2}

(d) none of these.

Q. If

α, β

be roots

x^{2} + p x + 1 = 0

and

γ, δ

are the roots of

x^{2} + q x + 1 = 0

, then

(a - γ) (β - γ) (α + δ) (β + δ)

Q. If

α, β

roots of

x^{2} + p x + 1 = 0

and

γ, δ

are the roots of

x^{2} + q x + 1 = 0

then prove that

(α - γ) (β - γ) (α + δ) (β + δ) = q^{2} - p^{2}

If ∝, β are the roots of $x^{2}$ + px + 1 = 0, γ, δ the roots of $x^{2}$ + qx + 1 = 0, then (∝ - γ)( β -γ)(∝+ δ)( β +δ) =

Q. lf

α, β

are the roots of

x^{2} + p x - q = 0

and

γ, δ

that of

x^{2} + p x + r = 0

, then

(α - γ) (β - γ) (α - δ) (β - δ) =

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If α,β are the roots of x2+px+1=0, δ,γ the roots of x2+qx+1=0, then the value of (α−γ)(β−γ)(α+δ(β+δ)) is

The correct option is C q2 - p2 α+β=−p , αβ=1 Also γ2+1=−qγ and δ2+1=−qδ (α−γ)(β−γ)(α+δ)(β+δ)=(αβ−(α+β)γ+γ2)(αβ+(α+β)δ+δ2)=(γ2+pγ+1)(δ2−pδ+1)=(pγ−qγ)(−pδ−qδ)=(q2−p2)γδ=(q2−p2)

If $α, β$ are the roots of $x^{2} + p x + 1 = 0,$ $δ, γ$ the roots of $x^{2} + q x + 1 = 0,$ then the value of $(α - γ) (β - γ) (α + δ (β + δ))$ is