Let `k` be sum of all x in the interval `[0, 2pi]` such that `3 cot^(2) x+8 cot x+3=0`, then the value of `k//pi` is ___________.

Answer

5

Explanation

`3cot^(2) x +8 cot x+3=0`
`:. cot x = (-4 pm sqrt(7))/3`
Both roots are negative.
`:. Pi/2 lt x_(1), x_(2) lt pi`
`:. Pi lt x_(1) + x_(2) lt 2pi`
Product of roots of roots `=cot x_(1) cot x_(2)=1`
`rArr cot x_(1) cot ((3pi)/2- x_(1))=1`
`rArr x_(1)+x_(2)=(3pi)/2`
Similarly, `x_(1)+x_(2)=(7pi)/2`