QJEEM20Ma3aff: JEE Main 2020 Mathematics - Class 11 JEEM20Ma3aff Solution: The natural number m, for which the coefficient of x in the…

Question

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The natural number m, for which the coefficient of x in the binomial expansion of

{\left( {{x^m} + {1 \over {{x^2}}}} \right)^{22}}

is 1540, is .............

Accepted Answer

General term,

{T_{r + 1}} = {}^{22}{C_r}{({x^m})^{22 - r}}{\left( {{1 \over {{x^2}}}} \right)^r} = {}^{22}{C_r}{x^{22m - mr - 2r}}

\because

{}^{22}{C_3} = {}^{22}{C_{19}} = 1540

\therefore

r = 3\,or\,19

22m - mr - 2r = 1

m = {{2r + 1} \over {22 - 5}}

When

r = 3

,

m = {7 \over {19}} \notin N

When

r = 19

,

m = {{38 + 1} \over {22 - 19}} = {{39} \over 3} = 13

\therefore

m = 13

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