QJEEM21Ma68e9: JEE Main 2021 Mathematics - Class 11 JEEM21Ma68e9 Solution: If the greatest value of the term independent of 'x' in the…

Question

Verified Solution • 2025–26 Session moderate

If the greatest value of the term independent of 'x' in the

expansion of

{\left( {x\sin \alpha + a{{\cos \alpha } \over x}} \right)^{10}}

is

{{10!} \over {{{(5!)}^2}}}

, then the value of 'a' is equal to :

Accepted Answer

{T_{r + 1}} = {}^{10}{C_r}{(x\sin \alpha )^{10 - r}}{\left( {{{a\cos \alpha } \over x}} \right)^r}

r = 0, 1, 2, ......., 10

T_{r + 1} will be independent of x when 10

-

2r = 0

\Rightarrow

r = 5

{T_6} = {}^{10}{C_5}{(x\sin \alpha )^5} \times {\left( {{{a\cos \alpha } \over x}} \right)^5}

= {}^{10}{C_5} \times {a^5} \times {1 \over {{2^5}}}{(\sin 2\alpha )^5}

will be greatest when sin2

\alpha

= 1

\Rightarrow {}^{10}{C_5}{{{a^5}} \over {{2^5}}} = {}^{10}{C_5} \Rightarrow a = 2

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