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QJEEM22Ma951b: JEE Main 2022 Mathematics - Class 11 JEEM22Ma951b Solution: If the sum of all the roots of the equation is , then p is…

If the sum of all the roots of the equation

e2x−11ex−45e−x+812=0{e^{2x}} - 11{e^x} - 45{e^{ - x}} + {{81} \over 2} = 0 is log⁡ep{\log _e}p , then p is equal to ____________.

Step-by-Step Verified Solution

Given that

e2x−11ex−45e−x+812=0e^{2 x}-11 e^x-45 e^{-x}+\frac{81}{2}=0

⇒2e3x−22e2x−90+81ex=0\Rightarrow 2 e^{3 x}-22 e^{2 x}-90+81 e^x=0

⇒2(ex)3−22(ex)2+81ex−90=0\Rightarrow 2\left(e^x\right)^3-22\left(e^x\right)^2+81 e^x-90=0

Let ex=ye^x=y

⇒2y3−22y2+81y−90=0\Rightarrow 2 y^3-22 y^2+81 y-90=0

Product of roots (y1,y2,y3)\left(y_1, y_2, y_3\right)

y1⋅y2⋅y3=−(−90)2=45y_1 \cdot y_2 \cdot y_3=\frac{-(-90)}{2}=45

Let x1,x2x_1, x_2 , and x3x_3 be roots of given equation

⇒ex1⋅ex2⋅ex3=45\Rightarrow e^{x_1} \cdot e^{x_2} \cdot e^{x_3} = 45

⇒ex1+x2+x3=45\Rightarrow e^{x_1+x_2+x_3} =45

⇒x1+x2+x3=log⁡e45=log⁡ep\Rightarrow x_1+x_2+x_3 =\log _e 45=\log _e p

⇒p=45\Rightarrow p = 45

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