The values of α and β such that equation x^2 + 2x + 2 + e^α – sinβ = 0 have a real solution is

Q : The values of α and β such that equation x² + 2x + 2 + e^α – sinβ = 0 have a real solution is

(A) $\large \alpha , \beta \in R $

(B) $\large \alpha \in (0 , 1) \; , \beta \in (\pi/2 , 2 \pi) $

(C) $\large \alpha \in (0 , \infty) \; and \beta \in (\pi/2 , \pi) $

(D) None of these

Ans: (D)

Solution: x² + 2x + 2 + e^α – sinβ = 0 has real roots if D ≥ 0

⇒  1 – 2 – e^α + sinβ  ≥ 0

⇒ sinβ  ≥  1 + e^α

Hence no real values of α and β are possible.