If the sum of all the roots of the equation e 2 x - 11 e x - 45 e - x + 81 2 = 0 is log e P , then P is equal to _____.

Given e 2 x - 11 e x - 45 e x + 81 2 = 0 e 3 x - 11 e 2 x + 81 2 e x - 45 = 0 Let e x = t , so t 3 - 11 t 2 + 81 2 t - 45 = 0 Product of roots t 1 t 2 t 3 = 45 i.e. e x 1 × e x 2 × e x 3 = 45 e x 1 + x 2 + x 3 = 45 x 1 + x 2 + x 3 = log 2 45 Given sum of root = ln P So P = 45

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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-Increasing-DecreasingHard2 minPYQ_2022

MathematicsHardnumerical

If the sum of all the roots of the equation $e^{2 x} - 11 e^{x} - 45 e^{- x} + \frac{81}{2} = 0$ is $\log_{e} P$ , then $P$ is equal to _____.

Answer:

45.00

Solution:

Given $e^{2 x} - 11 e^{x} - \frac{45}{e^{x}} + \frac{81}{2} = 0$

$e^{3 x} - 11 e^{2 x} + \frac{81}{2} e^{x} - 45 = 0$

Let $e^{x} = t$ , so $t^{3} - 11 t^{2} + \frac{81}{2} t - 45 = 0$

Product of roots $t_{1} t_{2} t_{3} = 45$

i.e. $e^{x_{1}} \times e^{x_{2}} \times e^{x_{3}} = 45$

$e^{x_{1} + x_{2} + x_{3}} = 45$

$x_{1} + x_{2} + x_{3} = \log_{2} 45$

Given sum of root $= \ln |P|$

So $P = 45$

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-Increasing-Decreasing

⏱ 2mℹ️ Source: PYQ_2022

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