QJEEM21Mae7c9: JEE Main 2021 Mathematics - Class 11 JEEM21Mae7c9 Solution: If is the term, independent of x, in the binomial expansion of…

Question

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If

\left( {{{{3^6}} \over {{4^4}}}} \right)k

is the term, independent of x, in the binomial expansion of

{\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}

, then k is equal to ___________.

Accepted Answer

{\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}

{T_{r + 1}} = {( - 1)^r}\,.\,{}^{12}{C_r}{\left( {{x \over 4}} \right)^{12 - r}}{\left( {{{12} \over {{x^2}}}} \right)^r}

{T_{r + 1}} = {( - 1)^r}\,.\,{}^{12}{C_r}{\left( {{1 \over 4}} \right)^{12 - r}}{\left( {12} \right)^r}\,.\,{(x)^{12 - 3r}}

Term independent of x

\Rightarrow

12

-

3r = 0

\Rightarrow

r = 4

{T_5} = {( - 1)^r}\,.\,{}^{12}{C_r}{\left( {{1 \over 4}} \right)^8}{\left( {12} \right)^4} = {{{3^6}} \over {{4^4}}}.\,k

\Rightarrow

k = 55

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