Decays of Bs and B0 mesons into J/psi pi +pi -pi +pi - final states, produced in pp collisions at the LHC, are investigated using data corresponding to an integrated luminosity of 3/fb collected with the LHCb detector. B^0_(s) -> Jpsi f_1(1285) decays are seen for the first time, and the branching fractions are measured. Using these rates, the f_1(1285) mixing angle between strange and non-strange components of its wave function in the q-qbar structure model is determined to be pm (24.0^{+3.1+0.6}_{-2.6-0.8}) degrees. Implications on the possible tetraquark nature of the f_1(1285) are discussed.
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