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A study of $B^0$~and $B^0_s$ meson decays into $J/\psi\eta$~and$J/\psi\eta^{\prime}$ final states is performed using a data set ofproton-proton collisions at centre-of-mass energies of 7 and 8TeV, collected bythe LHCb experiment and corresponding to 3.0fb$^{-1}$ of integrated luminosity.The decay $B^0 \rightarrow J/\psi \eta^{\prime}$ is observed for thefirst time.The following ratios of branching fractions are measured: $$\frac{\mathcal{B}(B^0 \rightarrow J/\psi \eta^{\prime})}{\mathcal{B}(B^0_s\rightarrow J/\psi \eta^{\prime})} =(2.28\pm0.65\,(stat)\,\pm0.10\,(syst)\,\pm0.13\,(f_{s}/f_{d}))\times10^{-2},$$$$ \frac{\mathcal{B}(B^0 \rightarrow J/\psi \eta)}{\mathcal{B}(B^0_s\rightarrow J/\psi \eta)} =(1.85\pm0.61\,(stat)\,\pm0.09\,(syst)\,\pm0.11\,(f_{s}/f_{d}))\times10^{-2},$$where the third uncertainty is related to the present knowledge of$f_{s}/f_{d}$, the ratio between the probabilities for a $b$ quark to form a$B^0_s$ or $B^0$ meson. The branching fraction ratios are used to determine theparameters of $\eta-\eta^{\prime}$ meson mixing. In addition, the firstevidence for the decay $B^0_s \rightarrow \psi(2S) \eta^{\prime}$ is reported,and the relative branching fraction is measured,  $$\frac{\mathcal{B}(B^0_s \rightarrow \psi(2S)\eta^{\prime})}{\mathcal{B}(B^0_s \rightarrow J/\psi \eta^{\prime})} =(38.7\pm9.0\,(stat)\,\pm1.3\,(syst)\,\pm0.9(\mathcal{B}))\times10^{-2},$$  where the third uncertainty is due to the limited knowledge of the branchingfractions of $J/\psi$ and $\psi(2S)$~mesons.

Study of $\eta-\eta^{\prime}$ mixing from measurement of $B^0_{(s)} \rightarrow J/\psi \eta^{(\prime)}$~decay rates