Question

Find the term independent of x in the expansion of `(3/2x^2-1/(3x))^6`.

Answer 1

`(r+1)^(th)` term in given expression,
`T_(r+1) = C(6,r)((3x^2)/2)^(6-r)(-1/(3x))^r`
`= C(6,r)(3/2)^(6-r)(-3)^(-r)(x^2)^((6-r)-r)`
`T_(r+1)= C(6,r)(3/2)^(6-r)(-3)^(-r)(x)^(12-3r)->(1)`
For `T_(r+1)` to be independent of `x`, `(12-3r) = 0`
`=>r = 4`
So, fifth term of given expression is independent of `x`.
`:. T_5 = C(6,4)(3/2)^2(-3)^(-4)` (putting value of r in (1))
` = 15**9/4**1/81 = 5/12`