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The coefficient of the term independent of $x$ in the expansion of $\bigg[\large\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-\sqrt x}\bigg]^{\normalsize 10}$ is ?

$\begin{array}{1 1} 110 \\ -105 \\ 70 \\ 210 \end{array} $

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Toolbox:
  • $(x^3+y^3)=(x+y)(x^2-xy+y^2)$
  • $x^2-y^2=(x+y)(x-y)$
$\large\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-\sqrt x}=\frac{(x^{1/3})^3+1^3}{x^{2/3}-x^{1/3}+1}-\frac{(\sqrt x)^2-1^2}{\sqrt x(\sqrt x-1)}$
$=\large\frac{(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)}{x^{2/3}-x^{1/3}+1}-\frac{(\sqrt x+1)(\sqrt x-1)}{\sqrt x(\sqrt x-1)}$
$=x^{1/3}+1-1-\large\frac{1}{\sqrt x}$
$=x^{1/3}-x^{-1/2}$
$\Rightarrow\:\bigg[\large\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-\sqrt x}\bigg]^{10}=$$\big(x^{1/3}-x^{-1/2}\big)^{10}$
General term in this expansion is
$(-1)^r.^{10}C_r.(x^{1/3})^{10-r}.(x^{-1/2})^r$
$=^{10}C_r.x^{(10-r)/3-r/2}$
For independent term in this expansion $\large\frac{10-r}{3}-\frac{r}{2}$$=0$
$\Rightarrow\:r=4$
Independent term = $^{10}C_4=210$
answered Oct 22, 2013 by rvidyagovindarajan_1
 

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