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Find the term independent of x in the ex...

Find the term independent of `x` in the expansion of `((3)/(2) x^(2) - (1)/(3x) )^(9)`.

Text Solution

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`T_(r+1)= ""^(9)C_(r) ((3)/(2) x^(2))^(9-r)( - (1)/(3x))^(r)`
` = (-1)^(r) . ""^(9)C_(r) ((3)/(2))^(9-r) ((1)/(3 ))^(r)x^(18-3r) " "`...(i)
If this term in indepandent f x then index of x must be
zero ,i.e. `18 - 3r = rArr r= 6`
Therefore , (r + 1) in term ,i.e., 7th term is independent f x
and its value by putting r = 6 in Eq .(i)
` = (-1)^(r) . ""^(9)C_(r) ((3)/(2))^(3) ((1)/(3 ))^(6)""^(9)C_(3) .(1)/(2^(3).3^(3)) `
` = (9*8*7)/((1*2*3)2^(3) .3^(3)) = (7)/(18)`
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