Observation of $B_c^{+}\to J\mathrm{/}\psi D^{(\ast )}{K}^{(\ast )}$ decays

A search for the decays $B_c^{+}\to J\mathrm{/}\psi D^{(\ast )0}{K}^{+}$ and $B_c^+ \to J/\psi D^{(*)+} K^{*0}$ is performed with data collected at the LHCb experiment corresponding to an integrated luminosity of 3 fb${}^{-1}$. The decays $B_c^+ \to J/\psi D^0 K^+$and$B_c^+ \to J/\psi D^{*0} K^+$ are observed for the first time, while first evidence is reported for the $B_c^{+}\to J\mathrm{/}\psi D^{\ast +}{K}^{\ast 0}$ and $B_c^{+}\to J\mathrm{/}\psi D^{+}{K}^{\ast 0}$ decays. The branching fractions of these decays are determined relative to the $B_c^{+}\to J\mathrm{/}\psi {\pi }^{+}$ decay. The $B_c^{+}$ mass is measured, using the $J\mathrm{/}\psi D^0{K}^{+}$ final state, to be $6274.28\pm 1.40(stat)\pm 0.32(syst)$ MeV/$c^2$. This is the most precise single measurement of the $B_c^{+}$ mass to date.