The term independent of \(x\) in the expansion of \(\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]^{10}, x \neq 1\), is equal to.....

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Binomial Theorem

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}, x \neq = 1

, is equal to.....

Answer: 210

Each fraction collapses. Treating

x^{1/3}

as the variable in the first,

\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

and in the second,

\frac{x - 1}{x - x ^{1/2}} = \frac{( x ^{1/2} - 1 ) ( x ^{1/2} + 1 )}{x ^{1/2} ( x ^{1/2} - 1 )} = 1 + x^{- 1/2}

so the bracket reduces to

x^{1/3} - x^{- 1/2}

and the expansion is

(x^{1/3} - x^{- 1/2})^{10}

. Its general term is

(r 10) (x^{1/3})^{10 - r} (- x^{- 1/2})^{r} = (r 10) (- 1)^{r} x^{\frac{10 - r}{3} - \frac{r}{2}}

and the power of

x

vanishes when

2 (10 - r) = 3 r

, that is

r = 4

. The term independent of

x

(4 10) (- 1)^{4} = 210

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Integer answer