JEE Main 2025 Apr 2 Shift 1, Mathematics Q6: Introduction to Binomial Expansion

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

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

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

General term,

$T_{r+1}={}^{10}{C}_r{\left(x^{\frac{1}{3}}\right)}^{10-r}{\left(-\frac{1}{\sqrt{x}}\right)}^r$

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

For the constant term,

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

$2(10-r)-3r=0$

$20-5r=0$

$\Rightarrow r=4$

$\therefore$ Constant term $={}^{10}{C}_4(-1{)}^4=210$

$\boxed{210}$