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QJEEM21Mafbb8: JEE Main 2021 Mathematics - Class 11 JEEM21Mafbb8 Solution: The number of real roots of the equation is :

The number of real roots of the equation e6x−e4x−2e3x−12e2x+ex+1=0{e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0 is :

Multiple Choice Options

  • A 2
  • B 4
  • C 6
  • D 1

Step-by-Step Verified Solution

e6x−e4x−2e3x−12e2x+ex+1=0{e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0

⇒(e3x−1)2−ex(e3x−1)=12e2x\Rightarrow {\left( {{e^{3x}} - 1} \right)^2} - {e^x}\left( {{e^{3x}} - 1} \right) = 12{e^{2x}}

(e3x−1)2(ex−e−x−e−2x)=12{\left( {{e^{3x}} - 1} \right)^2}\left( {{e^x} - {e^{ - x}} - {e^{ - 2x}}} \right) = 12

⇒ex−e−x−e−2x⏟increasing (let f(x))=12e3x−1⏟decreasing (let g(x))\Rightarrow \underbrace {{e^x} - {e^{ - x}} - {e^{ - 2x}}}_{increa{\mathop{\rm sing}\nolimits} \,(let\,f(x))} = {{12} \over {\underbrace {{e^{3x}} - 1}_{decrea{\mathop{\rm sing}\nolimits} \,(let\,g(x))}}}

JEE Main 2021 (Online) 25th July Morning Shift Mathematics - Quadratic Equation and Inequalities Question 87 English Explanation
⇒\Rightarrow No. of real roots = 2

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