QJEEM20Ma9f71: JEE Main 2020 Mathematics - Class 11 JEEM20Ma9f71 Solution: If the constant term in the binomial expansion of is 405, then…

Question

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If the constant term in the binomial expansion of

{\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}

is 405, then |k| equals :

Accepted Answer

{\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}

r^th term of the expansion,

T_r+1 = ¹⁰C_r

{\left( {\sqrt x } \right)^{10 - r}}{\left( {{{ - k} \over {{x^2}}}} \right)^r}

= ¹⁰C_r.

{x^{{{10 - r} \over 2}}}.{\left( { - k} \right)^r}.{x^{ - 2r}}

= ¹⁰C_r.

{x^{{{10 - 5r} \over 2}}}.{\left( { - k} \right)^r}

If it is constant term then

{{{10 - 5r} \over 2}}

= 0

\Rightarrow

r = 2

T₃ = ¹⁰C₂.(-k)² = 405

\Rightarrow

k² =

{{405} \over {45}}

= 9

\Rightarrow

k =

\pm

3

\Rightarrow

|k| = 3

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