If the constant term in the binomial expansion
of
${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$ is 405, then |k| equals :
Solution
${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$
<br><br>r<sup>th</sup> term of the expansion,
<br><br>T<sub>r+1</sub> = <sup>10</sup>C<sub>r</sub>${\left( {\sqrt x } \right)^{10 - r}}{\left( {{{ - k} \over {{x^2}}}} \right)^r}$
<br><br>= <sup>10</sup>C<sub>r</sub>.${x^{{{10 - r} \over 2}}}.{\left( { - k} \right)^r}.{x^{ - 2r}}$
<br><br>= <sup>10</sup>C<sub>r</sub>.${x^{{{10 - 5r} \over 2}}}.{\left( { - k} \right)^r}$
<br><br>If it is constant term then <br>${{{10 - 5r} \over 2}}$ = 0
<br>$\Rightarrow$ r = 2
<br><br>T<sub>3</sub> = <sup>10</sup>C<sub>2</sub>.(-k)<sup>2</sup> = 405
<br><br>$\Rightarrow$ k<sup>2</sup> = ${{405} \over {45}}$ = 9
<br><br>$\Rightarrow$ k = $\pm$ 3
<br><br>$\Rightarrow$ |k| = 3
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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