The term independent of x in the expression o | vedclass

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Question

$\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}$

General term of $\left(\frac{5}{2} x ^{3}-\frac{1}{5 x ^{2}}\ri

Vedclass · Accepted Answer

$\frac{33}{200}$

Vedclass · Answer

$\left(1-x^{2}+3 x^{3}ight)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}ight)^{11}$

General term of $\left(\frac{5}{2} x ^{3}-\frac{1}{5 x ^{2}}ight)^{11}$ is

${ }^{11} C_{r}\left(\frac{5}{2} x^{3}ight)^{11-r}\left(-\frac{1}{5 x^{2}}ight)^{ r }$

General term is ${ }^{11} C _{ r }\left(\frac{5}{2}ight)^{11- r }\left(-\frac{1}{5}ight)^{ r } x ^{33-5 r }$

Now, term independent of $x$

$1 	imes$ coefficient of $x^{0}$ in $\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}ight)^{11}$

$-1 	imes$ coefficient of $x^{-2}$ in $\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}ight)^{11}+$

$3 	imes$ coefficient of $x^{-3}$ in $\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}ight)^{11}$

$\begin{array}{ll}	ext { for coefficient of } x ^{0} & 33-5 r =0 	ext { not possible } \ 	ext { for coefficient of } x ^{-2} & 33-5 r =-2 \ & 35=5 r \Rightarrow r =7 \ 	ext { for coefficient of } x ^{-3} & 33-5 r =-3 \ & 36=5 r 	ext { not possible }\end{array}$

So term independent of $x$ is

$(-1)^{11} C_{7}\left(\frac{5}{2}ight)^{4}\left(-\frac{1}{5}ight)^{7}=\frac{33}{200}$

The term independent of $x$ in the expression of $\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}, x \neq 0$ is

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