Coefficient of the term independent of x in the expansion of x+1x2/3−x1/3+1−x−1x−x1/210is

We havex+1=x1/33+1=x1/3+1x2/3−x1/3+1,x−1=(x−1)(x+1)and x−x1/2=x(x−1),Thus, x+1x2/3−x1/3+1−x−1x−x1/2 =x1/3+1−1+x−1/2=x1/3−x−1/2Now, the (r+1)th term in the expansion of x1/3−x−1/210 is Tr+1=10Crx1/310−rx−1/2r(−1)r=10Crx103−5r/6(−1)rFor this term to be independent of x, 103−5r6=0 ⇒ r=103×65=4 Thus, coefficient of the term independent of x in the given expression is 10C4=10!4!6!=210.

Coefficient of the term independent of x in the expansion of x+1x2/3−x1/3+1−x−1x−x1/210is

We havex+1=x1/33+1=x1/3+1x2/3−x1/3+1,x−1=(x−1)(x+1)and x−x1/2=x(x−1),Thus, x+1x2/3−x1/3+1−x−1x−x1/2 =x1/3+1−1+x−1/2=x1/3−x−1/2Now, the (r+1)th term in the expansion of x1/3−x−1/210 is Tr+1=10Crx1/310−rx−1/2r(−1)r=10Crx103−5r/6(−1)rFor this term to be independent of x, 103−5r6=0 ⇒ r=103×65=4 Thus, coefficient of the term independent of x in the given expression is 10C4=10!4!6!=210.

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Coefficient of the term independent of $x$ in the expansion of ${(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}$ is

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$210$

$105$

$70$

$35$

answer is A.

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Detailed Solution

We have
$\begin{array}{l} x + 1 = {(x^{1 / 3})}^{3} + 1 = (x^{1 / 3} + 1) (x^{2 / 3} - x^{1 / 3} + 1), \\ x - 1 = (\sqrt{x} - 1) (\sqrt{x} + 1) \end{array}$
and $x - x^{1 / 2} = \sqrt{x} (\sqrt{x} - 1),$

Thus, $\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}}$
$= x^{1 / 3} + 1 - (1 + x^{- 1 / 2}) = x^{1 / 3} - x^{- 1 / 2}$
Now, the $(r + 1) t h$ term in the expansion of ${(x^{1 / 3} - x^{- 1 / 2})}^{10}$ is

$\begin{aligned} T_{r + 1} =^{10} C_{r} {(x^{1 / 3})}^{10 - r} {(x^{- 1 / 2})}^{r} (- 1)^{r} \\ =^{10} C_{r} x^{10 \sqrt{3} - 5 r / 6} (- 1)^{r} \end{aligned}$
For this term to be independent of $x$ ,
$\frac{10}{3} - \frac{5 r}{6} = 0 \Rightarrow r = \frac{10}{3} \times \frac{6}{5} = 4$
Thus, coefficient of the term independent of $x$ in the given expression is