QJEEM21Ma5db3: JEE Main 2021 Mathematics - Class 11 JEEM21Ma5db3 Solution: The sum of the roots of the equation , is :

Question

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The sum of the roots of the equation

x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0

, is :

Accepted Answer

x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0

{\log _2}({2^{x + 1}}) - {\log _2}{(3 + {2^x})^2} + {\log _2}(10 - {2^{ - x}}) = 0

lo{g_2}\left( {{{{2^{x + 1}}.(10 - {2^{ - x}})} \over {{{(3 + {2^x})}^2}}}} \right) = 0

{{2({{10.2}^{ - x}} - 1)} \over {{{(3 + {2^x})}^2}}} = 1

\Rightarrow {20.2^x} - 2 = 9 + {2^{2x}} + {6.2^x}

\therefore

{({2^x})^2} - 14({2^x}) + 11 = 0

Roots are 2^x₁ & 2^x₂

\therefore

2^x₁ . 2^x₂ = 11

x₁ + x₂ = log₂(11)

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