Mathematics - Binomial Theorem Question with Solution | TestHub

We know,$\left(x^{2/3}-x^{1/3}+1\right)\left(x^{1/3}+1\right)=x+1$

$\Rightarrow \frac{x+1}{x^{2/3}-x^{1/3}+1}=x^{1/3}+1$

Also,$\frac{x-1}{x-\sqrt{x}}=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}+1}{\sqrt{x}}=1+x^{-1/2}$

Hence, the given binomial becomes${\left(x^{1/3}-x^{-1/2}\right)}^{10}$.

General term${=}^{10}{C}_r{\left(-1\right)}^r{x}^{\frac{10-r}{3}}{x}^{-\frac{r}{2}}$

Hence, for term independent of$x.$

We get$\frac{10-r}{3}=\frac{r}{2}$

$20-2r=3r$

$r=4$
$\therefore$ Term Independent of $x{=}^{10}{C}_4=210$