The values of α and β for which the quadratic equation x2+2 x+2+e-2 α- cos β=0 has a real solution, is

Question

The values of α and β for which the quadratic equation x2+2 x+2+e-2 α- cos β=0 has a real solution, is (A) α, β ∈ R (B) α ∈(0,1), β ∈((π/2), 2 π) (C) α ∈(0, ∞), β ∈((π/2), π) (D) no real value of α, β possible

Tardigrade Admin · Accepted Answer

As the equation x 2+2 x +2+ e -2 a- cos β=0 text has real roots, so discriminant ≥ 0 ⇒ 4-4(2+ e -2 a- cos β) ≥ 0 ⇒ 1-2- e -2 α+ cos β ≥ 0 ⇒ cos β ≥ 1+ e 2 a, text which is not possible. So no real value of α, β text is possible.

Q. The values of $α$ and $β$ for which the quadratic equation $x^{2} + 2 x + 2 + e^{- 2 α} - cos β = 0$ has a real solution, is

$α, β \in R$

$α \in (0, 1), β \in (\frac{π}{2}, 2 π)$

$α \in (0, \infty), β \in (\frac{π}{2}, π)$

no real value of $α, β$ possible

As the equation
$x^{2} + 2 x + 2 + e^{- 2 a} - cos β = 0 has real roots, so discriminant \geq 0$
$\Rightarrow 4 - 4 (2 + e^{- 2 a} - cos β) \geq 0$
$\Rightarrow 1 - 2 - e^{- 2 α} + cos β \geq 0$
$\Rightarrow cos β \geq 1 + e^{2 a}, which is not possible. So no real value of α, β is possible.$

Q. The values of α and β for which the quadratic equation x2+2x+2+e−2α−cosβ=0 has a real solution, is

α,β∈R

α∈(0,1),β∈(2π​,2π)

α∈(0,∞),β∈(2π​,π)

no real value of α,β possible

As the equation x2+2x+2+e−2a−cosβ=0 has real roots, so discriminant ≥0 ⇒4−4(2+e−2a−cosβ)≥0 ⇒1−2−e−2α+cosβ≥0 ⇒cosβ≥1+e2a, which is not possible. So no real value of α,β is possible.

Q. The values of $α$ and $β$ for which the quadratic equation $x^{2} + 2 x + 2 + e^{- 2 α} - cos β = 0$ has a real solution, is

$α, β \in R$

$α \in (0, 1), β \in (\frac{π}{2}, 2 π)$

$α \in (0, \infty), β \in (\frac{π}{2}, π)$

no real value of $α, β$ possible