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Q.
The values of $\alpha$ and $\beta$ for which the quadratic equation $x^2+2 x+2+e^{-2 \alpha}-\cos \beta=0$ has a real solution, is
Complex Numbers and Quadratic Equations
Solution:
As the equation
$x ^2+2 x +2+ e ^{-2 a}-\cos \beta=0 \text { has real roots, so discriminant } \geq 0 $
$\Rightarrow 4-4\left(2+ e ^{-2 a}-\cos \beta\right) \geq 0 $
$\Rightarrow 1-2- e ^{-2 \alpha}+\cos \beta \geq 0 $
$\Rightarrow \cos \beta \geq 1+ e ^{2 a}, \text { which is not possible. So no real value of } \alpha, \beta \text { is possible. }$