Answer
Conservative
Work Step by Step
The vector field $F(x,y)=Pi+Qj$ a conservative field throughout the domain $D$, when
$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
Here, $P$ and $Q$ are the first-order partial derivatives on the domain $D$.
$\int_C \overrightarrow{F} \cdot \overrightarrow{dr}$ is independent of the path if and only if the line integral $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}=0$ for every closed curve $C$.
We know that the work done is the line integral of force. When we draw a closed loop around the center of the vector field, then we get a non-zero amount of work needed to get to the initial point.
For a conservative field, there will always be an equal amount of positive and negative work or zero work required to achieve the starting point.
This implies that the summation of these vectors gives the resultant vector with magnitude of $0$ and this can only be possible when the vector field represents a conservative field.
