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  • Find the coefficient of the term independent of x in the expansion of <img alt="\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}"

Find the coefficient of 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}

Option: 1

105


Option: 2

210


Option: 3

220


Option: 4

190


Answers (1)

best_answer

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

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

Let (r+1) th term be independent of x

\therefore \mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r} \cdot} \cdot(-1)^{\mathrm{r}}(\mathrm{x})^{\frac{10-\mathrm{r}}{3}}\left(\mathrm{x}^{-1 / 2}\right)^{\mathrm{r}}

The power of x is \frac{10-r}{3}-\frac{r}{2}=0 \Rightarrow r=4

Hencse the coefficient is ={ }^{10} C_4(-1)^4=\frac{10 !}{6 ! 4 !}=210

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Nehul

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