The term independent of x in the expansion of , is: [2025]

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Solution

Q.
The term independent of x in the expansion of ${(\frac{(x + 1)}{(x^{2 / 3} + 1 - x^{1 / 3})} - \frac{(x - 1)}{(x - x^{1 / 2})})}^{10}, x > 1$ , is: [2025]

1 120

2 240

3 210

4 150

Ans.
(3)

We have, ${(\frac{(x + 1)}{(x^{\frac{2}{3}} + 1 - x^{\frac{1}{3}})} - \frac{(x - 1)}{(x - x^{\frac{1}{2}})})}^{10}$

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

$T_{r + 1} = {}^{10}C_{r} {(x)}^{\frac{10 - r}{3}} {(- 1)}^{r} {(x)}^{- \frac{r}{2}}$

Since, the term is independent of x, then $\frac{10 - r}{3} - \frac{r}{2} = 0$

$\Rightarrow$ (20 – 2r) – 3r = 0 $\Rightarrow$ r = 4.

Hence, the required term is ${}^{10}C_{4} {(- 1)}^{4} = 210$ .

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