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4-2.Quadratic Equations and Inequations
hard
જો સમીકરણ $e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0$ નાં તમામ બીજનો સરવાળો $\log _{ e } P$હોય,તો$p=\dots\dots\dots$
A
$40$
B
$45$
C
$50$
D
$55$
(JEE MAIN-2022)
Solution
$\left.e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0\right]$
$\left.\left(e^{x}\right)^{3}-11\left(e^{x}\right)^{2}-45+\frac{81 e^{x}}{2}=0\right]$
$e^{x}=t$
$2 t^{3}-22 t^{2}+81 t-90=0$
$t_{1} t_{2} t_{3}=45$
$e^{x_{1}} \cdot e^{x_{2}} \cdot e^{x_{3}}=45$
$e^{x_{1}+x_{2}+x_{3}}=45$
$\log _{e} e^{x_{1}+x_{2}+x_{3}}=\log _{e} 45$
$x_{1}+x_{2}+x_{3}=\log _{e} 45$
$\log _{e} P=\log _{e} 45$
$P =45$
Standard 11
Mathematics
