We present a search for a new particle T′ decaying to top quark via T′→t+X, where X is an invisible particle. In a data sample with 4.8fb-1 of integrated luminosity collected by the CDF II detector at Fermilab in pp̄ collisions with √s=1.96TeV, we search for pair production of T′ in the lepton+jets channel, pp̄→tt̄+X+X→ νbqq′b+X+X. We interpret our results primarily in terms of a model where T′ are exotic fourth generation quarks and X are dark matter particles. Current direct and indirect bounds on such exotic quarks restrict their masses to be between 300 and 600GeV/c2, the dark matter particle mass being anywhere below Tm′. The data are consistent with standard model expectations, and we set 95% confidence level limits on the generic production of T ′T̄′→tt̄+X+X. For the dark matter model we exclude T′ at 95% confidence level up to Tm′=360GeV/c2 for mX≤100GeV/c2. © 2011 American Physical Society.