A search for the decays B+c→J/ψD(*)0K+ and B+c→J/ψD(*)+K*0 is performed with data collected at the LHCb experiment corresponding to an integrated luminosity of 3  fb−1. The decays B+c→J/ψD0K+ and B+c→J/ψD*0K+ are observed for the first time, while first evidence is reported for the B+c→J/ψD*+K*0 and B+c→J/ψD+K*0 decays. The branching fractions of these decays are determined relative to the B+c→J/ψπ+ decay. The B+c 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 B+c mass to date.