Mathematics - Binomial Theorem Question with Solution | TestHub

${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}$

${\left(\frac{\left(x^{1/3}+1\right)\left(x^{2/3}-x^{1/3}+1\right)}{x^{2/3}-x^{1/3}+1}-\frac{\left(x^{1/2}+1\right)\left(x^{1/2}-1\right)}{x^{1/2}\left(x^{1/2}-1\right)}\right)}^{10}$

$={\left(\left(x^{1/3}+1\right)-\left(\frac{x^{1/2}+1}{x^{1/2}}\right)\right)}^{10}$

$={\left(x^{1/3}-\frac{1}{x^{1/2}}\right)}^{10}$

Now the ${\left(r+1\right)}^{\mathrm{th}}$ Term

$T_{r+1}={}^{10}C_r{\left(x^{1/3}\right)}^{10-r}·{\left(-\frac{1}{x^{1/2}}\right)}^r$

$T_{r+1}={}^{10}C_r{\left(x^{1/3}\right)}^{10-r}·x^{-r/2}{\left(-1\right)}^r$

For independent term

$\frac{10-r}{3}-\frac{r}{2}=0\Rightarrow r=4$

$\Rightarrow T_5={}^{10}C_4{\left(-1\right)}^4$

$=\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}=210$