Let α, β be the roots of x2-x+p=0 and γ, δ be the roots of x2-4 x+q=0. If α, β, γ, δ are in G.P., then the integral values of p and q respectively, are

Question

Let α, β be the roots of x2-x+p=0 and &gamma;, δ be the roots of x2-4 x+q=0. If α, β, &gamma;, δ are in G.P., then the integral values of p and q respectively, are (A) -2,-32 (B) -2,3 (C) -6,3 (D) -6,-32

Tardigrade Admin · Accepted Answer

text Let β=α r , &gamma;=α r 2, δ=α r 3 α+β=1 ⇒ α(1+ r )=1 &gamma;+δ=4 ⇒ α(1+ r ) r 2=4 ⇒ r 2=4 ⇒ r = ± 2 r =2 ⇒ α=(1/3), b =(2/3) text and p text is not an integer. ∴ r =-2, α=-1, β=2, &gamma;=-4, δ=8 p =α ⋅ β=-2, q =&gamma; ⋅ δ=-32

Q. Let $α, β$ be the roots of $x^{2} - x + p = 0$ and $γ, δ$ be the roots of $x^{2} - 4 x + q = 0$ . If $α, β, γ, δ$ are in G.P., then the integral values of $p$ and $q$ respectively, are

$- 2, - 32$

$- 2, 3$

$- 6, 3$

$- 6, - 32$

Q. Let α,β be the roots of x2−x+p=0 and γ,δ be the roots of x2−4x+q=0. If α,β,γ,δ are in G.P., then the integral values of p and q respectively, are

−2,−32

−2,3

−6,3

−6,−32

Let β=αr,γ=αr2,δ=αr3 α+β=1⇒α(1+r)=1 γ+δ=4⇒α(1+r)r2=4⇒r2=4⇒r=±2 r=2⇒α=31​,b=32​ and p is not an integer. ∴r=−2,α=−1,β=2,γ=−4,δ=8 p=α⋅β=−2,q=γ⋅δ=−32

Q. Let $α, β$ be the roots of $x^{2} - x + p = 0$ and $γ, δ$ be the roots of $x^{2} - 4 x + q = 0$ . If $α, β, γ, δ$ are in G.P., then the integral values of $p$ and $q$ respectively, are

$- 2, - 32$

$- 2, 3$

$- 6, 3$

$- 6, - 32$