Recall Maxwell's equations:
$\begin{align*}\nabla \cdot \mathbf{E} &= \frac{1}{\epsilon_0}\rho \\\nabla \times \mathbf{E} &= -\frac{\partial \math...$
If you plug the given electric field into Gauss's Law, you will find that
$\begin{align*}k = \frac{1}{\epsilon_0}\rho\end{align*}$
We are told that $k$ is a nonzero constant. Therefore, $\rho$ must also be nonzero, and so there must be charge density in the region. Therefore, answer (B) is correct.

Solution to 1992 Problem 64