If the term free from `x` in the expansion of `(sqrt(x)-k/(x^2))^(10)` is `405` , find the value of `kdot`

Explanation

Give expansion is `(sqrtx - (k)/(x^(2)))^(10)`
Let `T_(r + 1)` is the general term,
Then, `T_(r + 1) = .^(10)C_(r) (sqrtx)^(10 - r) ((-k)/(x^(2)))^(r)`
`= .^(10)C_(r) (x)^((1)/(2) (10 - r)) (-k)^(r). S^(-2r)`
`= .^(10)C_(r) x^(5 - (r)/(2)) (-k)^(r) .x^(-2r)`
`= .^(10)C_(r) x^(5 - (r)/(2) - 2r) (-k)^(r)`
`= .^(10)C_(r) x^(10 - 5r)/(2) (-k)^(r)`
For free from x, `(10 - 5r)/(2) = 0`
`rArr 10 - 5r = 0 rArr r = 2`
Since, `T_(2 + 1) = T_(3)` is free from x.
`:. T_(2 + 1) = .^(10)C_(2) (-k)^(2) = 405`
`rArr 45 k^(2) = 405 rArr k^(2) = (405)/(45) = 9`
`:. k = -+ 3`