The term independent of x in the expansion of | vedclass

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Question

$\left.\left( x ^{1 / 3}+1\right)-\left(\frac{\sqrt{ x }+1}{\sqrt{ x }}\right)\right)^{10}$

$\left( x ^{1 / 3}- x ^{-1 / 2}\right)^{10}$

$T _{ r

Vedclass · Accepted Answer

$210$

Vedclass · Answer

$\left.\left( x ^{1 / 3}+1ight)-\left(\frac{\sqrt{ x }+1}{\sqrt{ x }}ight)ight)^{10}$

$\left( x ^{1 / 3}- x ^{-1 / 2}ight)^{10}$

$T _{ r +1}={ }^{10} C _{ r }\left( x ^{1 / 3}ight)^{10- r }\left(- x ^{-1 / 2}ight)^{ r }$

$\frac{10- r }{3}-\frac{ r }{2}=0 \Rightarrow 20-2 r -3 r =0$

$\Rightarrow r =4$

$T _{5}={ }^{10} C _{4}=\frac{10 	imes 9 	imes 8 	imes 7}{4 	imes 3 	imes 2 	imes 1}=210$

The term independent of $x$ in the expansion of $\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1,$ is equal to ....... .

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