A search for the decays Bc+→J/ψD(∗)0K+ and Bc+→J/ψD(∗)+K∗0 is performed with data collected at the LHCb experiment corresponding to an integrated luminosity of 3 fb-1. The decays Bc+→J/ψD0K+ and Bc+→J/ψD∗0K+ are observed for the first time, while first evidence is reported for the Bc+→J/ψD∗+K∗0 and Bc+→J/ψD+K∗0 decays. The branching fractions of these decays are determined relative to the Bc+→J/ψπ+ decay. The Bc+ mass is measured, using the J/ψD0K+ final state, to be 6274.28±1.40(stat)±0.32(syst) MeV/c2. This is the most precise single measurement of the Bc+ mass to date.