  We study the combined effect of a conical topological defect and a Coulomb
charge impurity on the dynamics of Dirac fermions in gapped graphene. Beyond a
certain strength of the Coulomb charge, quantum instability sets in, which
demarcates the boundary between sub and supercritical values of the charge. In
the subcritical regime, for certain values of the system parameters, the
allowed boundary conditions in gapped graphene cone can be classified in terms
of a single real parameter. We show that the observables such as local density
of states, scattering phase shifts and the bound state spectra are sensitive to
the value of this real parameter, which is interesting from an empirical point
of view. For a supercritical Coulomb charge, we analyze the system with a
regularized potential as well as with a zigzag boundary condition and find the
effect of the sample topology on the observable features of the system.
