In the expansion of `(x^3-1/(x^2))^n ,n in N` , if the sum of the coefficients of `x^5a n dx^(10)` is zero , then `n` is a. 25 b. 20 c. 15 d. none of these

Options

A. 25
B. 20
C. 15
D. None of these

Answer

C

Explanation

`(1/(x^(2)) - x^(3))^(n)`
General term `T_(r+1) = (n!)/(r!(n-r)!) (-1)^(n-r)x^(5r-2n)`
If `5r-2n=5`, then `5r = 2n+5` or `r = (2n)/(5) + 1`
If `5r - 2n = 10`, then `5r = 2n+10` or `r = (2n)/(5) + 2`
`x^(5)` and `x^(10)` terms occurs if `n = 5k`
Given that sum of `x^(5)` and `x^(10)` is zero.
`rArr (5k!)/((2k+1)!(3k-1)!) - (5k!)/((2k+2)!(3k-2)!) = 0`
or `(1)/(3k-1) - (1)/(2k+2) =0`
or `k = 3 rArr n = 15`