Question:
The term independent of ' $x$ ' in the expansion of
$\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}$, where $x \neq 0,1$ is equal
to_________.
Solution:
$\left(\left(x^{1 / 3}+1\right)-\left(\frac{x^{1 / 2}+1}{x^{1 / 2}}\right)\right)^{10}$
$=\left(x^{1 / 3}-\frac{1}{x^{1 / 2}}\right)^{10}$
Now General Term
$\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{1 / 3}\right)^{10-\mathrm{r}} \cdot\left(-\frac{1}{\mathrm{x}^{1 / 2}}\right)^{\mathrm{r}}$
For independent term
$\frac{10-r}{3}-\frac{r}{2}=0 \Rightarrow r=4$
$\Rightarrow \mathrm{T}_{5}={ }^{10} \mathrm{C}_{4}=210$
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