Find the term independent of x in the expansion of (x + 1/x raised to the power of 2/3 - x raised to the power of 1/3 + 1) - (x - 1/x raised to the power of 2/3 + x raised to the power of 1/3 + 1) raised to the power of 10.

B. 210 — Simplify the expression using algebraic identities to isolate the binomial expansion, then solve for the term where the power of x is zero.

JEE Main · MathematicsMedium

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

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Correct answer: B — $210$

$Simplify the first term using the sum of cubes identity: \frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} = \frac{( x ^{1/3} ) ^{3} + 1 ^{3}}{x ^{2/3} - x ^{1/3} + 1} = x^{1/3} + 1 .$
$Simplify the second term: \frac{x - 1}{x - x ^{1/2}} = \frac{( x - 1 ) ( x + 1 )}{x ( x - 1 )} = \frac{x + 1}{x} = 1 + x^{- 1/2} .$
$Substitute these into the original expression to get (x^{1/3} + 1 - (1 + x^{- 1/2}))^{10} = (x^{1/3} - x^{- 1/2})^{10} .$
$The general term is T_{r + 1} = (r 10) (x^{1/3})^{10 - r} (- x^{- 1/2})^{r} = (r 10) (- 1)^{r} x^{(10 - r) /3 - r /2} .$
$Set the exponent to zero: \frac{10 - r}{3} - \frac{r}{2} = 0, which simplifies to $20 - 2r - 3r = 0$, giving r = 4 .$
$Calculate the term: (4 10) = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 . Hence the answer is (B).$