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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`

A

4

B

120

C

210

D

310

Text Solution

Verified by Experts

`[(x+1)/(x^2//3)-x^(1//3)+1)-((x-1)/(x-x^(1//2))]^(1-)`
`= [((x^(1//3))^3+1^3)/(x^23-x^(1//3)+1)-({(sqrt(x))^2-1})/(sqrt(x)(sqrtx-1)]]^10`
` =[((x^(1//3)+1)(x^(2//3)+1-x^(1//3)))/(x^(23)-x^(1//3)+1)-({(sqrt(x))^2-1})/(sqrt(x)(sqrt(x)-1))]^10`
` therefore` The general term is
For independent of x, put
`(10-r)/(3)-(r)/(2)=0rArr 20-2r-3r=0`
`rArr 20=5rrArr r=4`
` therefore T_5=.^10C_4=(10xx9xx8xx7)/(4xx3xx2xx1)=210`
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