Abstract: The product of the A(b)(0) ((B) over bar (0)) differential production cross-section and the branching fraction of the decay A(b)(0)-> J/psi pK(-) ((B) over bar (0)-> J/psi p (K) over bar*(892)(0)) is measured as a function of the beauty hadron transverse momentum, p(T), and rapidity, y. The kinematic region of the measurements is p(T) <20 GeV/c and 2.0 < y < 4.5. The measurements use a data sample corresponding to an integrated luminosity of 3fb(-1) collected by the LHCb detector in pp collisions at centre-of-mass energies root s=7 TeV in 2011 and root s=8 TeV in 2012. Based on previous LHCb results of the fragmentation fraction ratio, f(Ab0)/f(d), the branching fraction of the decay A(b)(0)-> J/psi pK(-) is measured to be B(A(b)(0)-> J/psi pK(-))=(3.17 +/- 0.04 +/- 0.07 +/- 0.34(-0.28)(+0.45))x10(-4) where the first uncertainty is statistical, the second is systematic, the third is due to the uncertainty on the branching fraction of the decay (B) over bar (0)-> J/psi p (K) over bar*(892)(0), and the fourth is due to the knowledge of f(Ab0)/f(d). The sum of the asymmetries in the production and decay between A(b)(0) and (A) over bar (0)(b) is also measured as a function of p(T) and y. The previously published branching fraction of A(b)(0)-> J/psi p pi(-), relative to that of A(b)(0)-> J/psi pK(-), is updated. The branching fractions of A(b)(0)-> P-c(+)(-> J/psi p)K- are determined.