A search for the decays B-c(+) -> J/psi D-(*()0) Kappa(+) and B-c(+) -> 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(+) -> J/psi(DK+)-K-0 and B-c(+) -> J/psi D*K-0(+) are observed for the first time, while first evidence is reported for the B-c(broken vertical bar) -> J/psi D*K-broken vertical bar(*0) and B-c(broken vertical bar) -> J/psi(DK)-K-broken vertical bar*(0) decays. The branching fractions of these decays are determined relative to the B-c(+) -> J/psi pi(+) decay. The B-c(+) mass is measured, using the J/psi(DK+)-K-0 final state, to be 6274.28 +/- 1.40(stat) +/- 0.32(syst) MeV/c(2). This is the most precise single measurement of the B-c(+) mass to date.
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