Find the term independent of x in the expansion of :
(i) `(2x+1/(3x^(2)))^(9)` (ii) `((3x^(2))/2-1/(3x))^(6)`
(iii) `(x-1/x^(2))^(3n)` (iv) `(3x-2/x^(2))^(15)`

Answer

(i)`1792/9` (ii) `5/12` (iii) `(-1)^(n)*.^(3n)C_(n)` (iv)`-3003 xx3^(10) xx 2^(5)`

Explanation

(iii) `T_(r+1)= (-1)^(r)* ^(3n)C_(r)x^((3n-r))xx(1/x^(2))^(r)`
` rArr T_(r+1) = (-1)^(r)*^(3n)C_(r)x^((3n-3r)).`
Put `3n-3r = 0 rArr r = n.`
So, `T_(n+1)` is free from x.
And, `T_(n+1) = (-1)^(n)*^(3n)C_(r)x^((3n-3r)).`