the term independent of ' \(x\) ' in the expression of \(\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}\), where \(x \neq 0,1\) is equal to ____

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Binomial Theorem

the term independent of '

x

' in the expression of

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

, where

x \neq = 0, 1

is equal to ____

Answer: 210

Simplify each fraction:

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

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

Thus the expression becomes:

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

The general term in the binomial expansion is:

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

For the term independent of

x

, set the exponent to zero:

\frac{10 - k}{3} - \frac{k}{2} = 0 \Rightarrow 2 (10 - k) - 3 k = 0 \Rightarrow 20 = 5 k \Rightarrow k = 4

The coefficient is

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

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Integer answer