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4-2.Quadratic Equations and Inequations
hard
સમીકરણ $e^{4 x}+4 e^{3 x}-58 e^{2 x}+4 e^{x}+1=0$ નાં વાસ્તવિક ઉંકેલોની સંખ્યા..........
A
$6$
B
$9$
C
$20$
D
$2$
(JEE MAIN-2022)
Solution
$e^{4 x}+4 e^{3 x}-58 e^{2 x}+4 e^{x}+1=0$
Let $f(x)=e^{2 x}\left(e^{2 x}+\frac{1}{e^{2 x}}+4\left(e^{x}+\frac{1}{e^{x}}\right)-58\right)$
$e^{x}+\frac{1}{e^{x}}$
Let $h(t)=t^{2}+4 t-58=0$
$t =\frac{-4 \pm \sqrt{16+4.58}}{2}$
$\frac{-4 \pm 2 \sqrt{62}}{2}$
$t _{1}=-2+2 \sqrt{62}$
$t _{2}=-2-2 \sqrt{62}$ (not possible)
$t \geq 2$
$e ^{ x }+\frac{1}{ e ^{ x }}=-2+2 \sqrt{62}$
$e ^{2 x }-(-2+2 \sqrt{62}) e ^{ x }+1=0$
$(-2+2 \sqrt{62})-4$
$4+4.62-8 \sqrt{62}-4$
$248-8 \sqrt{62}>0$
$\frac{- b }{2 a }>0$
both roots are positive
$2$ real roots
Standard 11
Mathematics
