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Find the cofficient of the term independ...

Find the cofficient of the term independent of x in the expansion of `((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^1/2))^10`

Text Solution

Verified by Experts

The correct Answer is:
C
`[(x+1)/(x^(2//3)-x^(1//3)+1)-(x-1)/(x-x^(1//2))]^(10)`
`= [((x^(1//3)+1)(x^(2//3)-x^(1//3)+1))/((x^(2//3)-x^(1//3)+1))-((x^(1//2)-1)(x^(1//2) + 1))/(x^(1//2)(x^(1/2)-1))]^(10)`
`( :' x + 1= (x^(1//3))^(3)+1^(3))`
`= [(x^(1//3) + 1) - (1+x^(-1//2))]^(10) = [x^(1//3) - x^(-1//2)]^(10)`
`rArr T_(r+1) = .^(10)C_(r)(x^(1//3))^(10-r) (-x^(-1//2))^(r)`
`rArr T_(r+1) = .^(10)C_(r)x^((10-r)/(3).(r)/(2)).(-1)^(r)`
`rArr T_(r+1) = .^(10)C_(r)x^((20-5r)/(6)).(-1)^(r)`
For constant term,
`20- 5r = 0`
`rArr r= 4`
`rArrT_(5) = .^(10)C_(4) = 210`
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