QJEEM21Ma153d: JEE Main 2021 Mathematics - Class 11 JEEM21Ma153d Solution: If satisfies the equation t 2 - 9t + 8 = 0, then the value of…

Question

Verified Solution • 2025–26 Session moderate

If

{e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}

satisfies the equation t² - 9t + 8 = 0, then the value of

{{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 &lt; x &lt; {\pi \over 2}} \right)

is :

Accepted Answer

{e^{({{\cos }^2}x + {{\cos }^4}x + ...........\infty )\ln 2}} = {2^{{{\cos }^2}x + {{\cos }^4}x + ...........\infty }}

=

{2^{{{{{\cos }^2}x} \over {1 - {{\cos }^2}x}}}}

= {2^{{{\cot }^2}x}}

Given,

{t^2} - 9t + 8 = 0 \Rightarrow t = 1,8

\Rightarrow {2^{{{\cot }^2}x}} = 1,8 \Rightarrow co{t^2}x = 0,3

0 &lt; x &lt; {\pi \over 2} \Rightarrow \cot x = \sqrt 3

\therefore

{{2\sin x} \over {\sin x + \sqrt 3 \cos x}} = {2 \over {1 + \sqrt 3 \cot x}} = {2 \over 4} = {1 \over 2}

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