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Measurements of the normalized rapidity (y) and transverse momentum (qT) distributions of Drell-Yan muon and electron pairs in the Z-boson mass region (60<M(ll)<120 GeV) are reported. The results are obtained using a data sample of proton-proton collisions at a center-of-mass energy of 7 TeV, collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of 36 inverse picobarns. The distributions are measured over the ranges |y|<3.5 and qT<600 GeV and compared with QCD calculations using recent parton distribution functions. Overall agreement is observed between the models and data for the rapidity distribution, while no single model describes the Z transverse-momentum distribution over the full range.