1.

Theterm independent ofx ( x gt 0,xne1 )intheexpansionof[ ((x+ 1 ))/((x ^(2//3) -x^(1//3 ) + 1 )) -(( x - 1 )) /(( x- sqrt x )) ] ^(10)is

Answer»

`105 `
`210`
315
420

Solution : `[ ((x+ 1 ))/((x ^(2//3) - x ^(1//3)+ 1 )) - ((x - 1 ))/(x - SQRT x )]^(10)`
`= [ ((x ^(1/3)) ^3+1 ^(3))/( (x ^(2//3) -x ^(1/3)+ 1 ) )- ((sqrt x- 1 )(sqrt x+ 1 ))/(sqrtx (sqrt x - 1 ))] `
` [ (x ^(1/3) + 1 )-((sqrtx+ 1 ))/(sqrtx)]^(10)= [ x ^(1/3)-x ^( -1/2 ) ]^(10) `
LET` T _( r + 1 )`be independentof`.x.`
` rArrT_ (r + 1 )= ""^(10) c _r (x ^(1/3 )^(10- r )(-x ^(-1/2 ) ^ r`
`x ^( (10- r ) / (3) ) .x ^(- (r)/(2))=x ^(0) `
` rArr(10)/(3)-(r)/(3) -(r)/(2)=0 `
` rArr(10)/(3) = ( 5r ) /(6 ) `
` rArr r=4 `
`therefore `Thetermsis ` ""10 c_4( -1) ^(4)`
`= 210 `


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