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

detailed solution

Correct option is A

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.