The decay B0 -> D K*0 and the charge conjugate mode are studied using 1.0fb-1 of pp collision data collected by the LHCb experiment at sqrt(s) = 7TeV in 2011. The CP asymmetry between the B0 -> D K*0 and the B0bar -> D K*0bar decay rates, with the neutral D meson in the CP-even final state K+K-, is found to be A_d_KK = -0.45 +- 0.23 +- 0.02, where the first uncertainty is statistical and the second is systematic. In addition, favoured B0 -> D K*0 decays are reconstructed with the D meson in the non-CP eigenstate K+ pi-. The ratio of the B-flavour averaged decay rates in D decays to CP and non-CP eigenstates is measured to be R_d_KK = 1.36 (+0.37) (-0.32) +- 0.07, where the ratio of the branching fractions of D0 -> K- pi+ to D0 -> K+ K- decays is included as multiplicative factor. The CP asymmetries measured with two control channels, the favoured B0 -> DK*0 decay with D -> K+ pi- and the Bs0bar -> D K*0 decay with D K+ K-, are also reported.
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