Which of the following is/are conservative force(s)?

Answer

A, B, C

Explanation

Since , `W = int vec(F), vec(dr)`
Clearly for forces (A) and (B) the integration do not require any information of the path taken.
Force (C) :
`W_(c) = int (3 (xhat(i) +y hatj))/((x^(2)+y^(2))^(3//2)) (dx hat(i)+dy hat(j))`
`=3 int (x dx +y dy)/((x^(2)+y^(2))^(3//2))`
Taking : `x_(2) + y_(2) =t`
`2x dx + 2y dy = dt`
`rArr xdx + ydy = (dt)/(2)`
`rArr W_(c) = 3 int (dt//2)/(t^(3//2))=(3)/(2) int (dt)/(t^(3//2))`
which is solvable.
Hence (A), (B) and (C) are conservative forces
But (D) requires some more information on path. Hence non-conservative.