If all the roots of the equation x3-3 x=0 satisfy the equation (α- sin -1( sin 2)) x2-(β- tan -1( tan 1)) x+γ2-2 γ+1=0 , then find the value of | cot (β+γ)+ cot α|.

Question

If all the roots of the equation x3-3 x=0 satisfy the equation (α- sin -1( sin 2)) x2-(β- tan -1( tan 1)) x+&gamma;2-2 &gamma;+1=0 , then find the value of | cot (β+&gamma;)+ cot α|. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

∵ x=0, √3,-√3 satisfy the Q.E. ∴ It is an identity ∴ α- sin -1( sin 2)=0 ⇒ α= sin -1 sin 2=π-2 β= tan -1 tan 1=1 &gamma;2-2 &gamma;+1=0 ⇒ &gamma;=1 ∴ E=| cot (β+&gamma;)+ cot α| =| cot 2+ cot (π-2)|=0

Q. If all the roots of the equation $x^{3} - 3 x = 0$ satisfy the equation $(α - sin^{- 1} (sin 2)) x^{2} - (β - tan^{- 1} (tan 1)) x + γ^{2} - 2 γ + 1 = 0$ , then find the value of $∣ cot (β + γ) + cot α ∣$ .

$∵ x = 0, 3, - 3$ satisfy the $Q . E$ .
$∴$ It is an identity
$∴ α - sin^{- 1} (sin 2) = 0$
$\Rightarrow α = sin^{- 1} sin 2 = π - 2$
$β = tan^{- 1} tan 1 = 1$
$γ^{2} - 2 γ + 1 = 0 \Rightarrow γ = 1$
$∴ E = ∣ cot (β + γ) + cot α ∣$
$= ∣ cot 2 + cot (π - 2) ∣ = 0$

Q. If all the roots of the equation x3−3x=0 satisfy the equation (α−sin−1(sin2))x2−(β−tan−1(tan1))x+γ2−2γ+1=0 , then find the value of ∣cot(β+γ)+cotα∣.

∵x=0,3​,−3​ satisfy the Q.E. ∴ It is an identity ∴α−sin−1(sin2)=0 ⇒α=sin−1sin2=π−2 β=tan−1tan1=1 γ2−2γ+1=0⇒γ=1 ∴E=∣cot(β+γ)+cotα∣ =∣cot2+cot(π−2)∣=0

Q. If all the roots of the equation $x^{3} - 3 x = 0$ satisfy the equation $(α - sin^{- 1} (sin 2)) x^{2} - (β - tan^{- 1} (tan 1)) x + γ^{2} - 2 γ + 1 = 0$ , then find the value of $∣ cot (β + γ) + cot α ∣$ .

$∵ x = 0, 3, - 3$ satisfy the $Q . E$ .
$∴$ It is an identity
$∴ α - sin^{- 1} (sin 2) = 0$
$\Rightarrow α = sin^{- 1} sin 2 = π - 2$
$β = tan^{- 1} tan 1 = 1$
$γ^{2} - 2 γ + 1 = 0 \Rightarrow γ = 1$
$∴ E = ∣ cot (β + γ) + cot α ∣$
$= ∣ cot 2 + cot (π - 2) ∣ = 0$