Answer
The vector field $\overrightarrow{F}$ is not conservative.
Work Step by Step
When $F(x,y)=Ai+Bj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial A}{\partial y}=\dfrac{\partial B}{\partial x}$
$a$ and $b$ are the first-order partial derivatives on the domain $D$.
$\int_C \overrightarrow{F} \cdot \overrightarrow{dr}$ is independent of path if and only if $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}=0$ for every closed curve $C$.
The work done is a line integral of force.
We are given that the work done $\overrightarrow{F}$ along two differnet paths $C_1$ and $C_2$ that joins the same two points is different.
This means that the line integral of $\overrightarrow{F}$ is not path independent.
Hence, the vector field $\overrightarrow{F}$ is not conservative.
