1.

Ifabne0andthe sum ofcoefficientsofx ^ 7 andx ^ 4inthe expansionof((x ^ 2 ) /(a)-(b)/(x) ) ^ (11)is0 , then

Answer»

`a=B `
`a+ b= 0 `
`ab =- 1 `
`a b = 1 `

Solution :Consider` T_(r + 1 ) `termintheexpansion
`therefore T _(r + 1 )=""^(11)c_r ((X^ 2 )/(a)) ^(11 -r )(( - b ) /( x ) ) ^r`
`=""^(11) c _r ((1 )/(a)) ^( 11 - r)( - b) ^rx ^( 22 - 2 r) .x^( - r) `
`=""^(11)c _ r((1 ) /(a)) ^(11 - r )( - b ) ^rx ^(22 -3r ) `
if `x ^(22 - 3r)=x ^7`
`rArr22 - 3r=7`
`rArr3r=15`
`thereforer=5`
If`x^( 22-3r )=x ^ 4`
`rArr22 -3r= 4 `
`3r =18 `
`r=6 `
`therefore`co -efficientof` T_ 6`+ Co-efficientof`T _ 7=0 `
`rArr""^(11)c _ 6 (( 1 ) /(a)) ^ 5(- b ) ^( 6)+""^(11)c _ 5 (( 1 ) /(a)) ^6(- b ) ^( 5)=0 `
`rArr ""^(11)c _5[ (b^6 )/(a ^5 )-(b^5 ) /(a^6 ) ] =0""[ because""^(11)c _ 5 =""^(11)c _6] `
`rArr(b ^ 6 ) /( a ^5 )=(b ^ 5 ) /(a ^6 ) `
` rArr(ab) ^6 =(ab) ^ 5 `
` rArr(ab) ^5 [ab -1 ] =0 `
` rArrab - 1=0[becauseab ne 0] `
`ab = 1`


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