Non-relativistic energy levels, binding energies and oscillator strengths of CdSe⁄ZnT e core/shell spherical quantum dots with or without impurities submitted to an external magnetic field have been investigated by using a B-spline based variational method, within the framework of the effective mass approximation. In the case where the system contains hydrogenic impurity, the effects of its off-center displacement combined to the height of the confining potential have also been studied. The dielectric constant as well as the effective mass are considered to be dependent on the radius. The modifications occurring due to the presence of the magnetic field have been analyzed. We have found that the electronic and optical properties are strongly affected by the magnetic field strength, the spatial confinement and the off-center displacement. The oscillator strengths and the binding energies increase with the magnetic field, but its effect on the binding energies (both ground and excited states) is dimmed by the reduction of the core-to-shell radii ratio and the increase of the off-center displacement to the vicinity of the shell. However, the oscillator strengths increase with the magnetic field when the off-center displacement increases towards the shell.