We present a new prefixed tableau system TK for verification of validity in modal logic K. The system TK is deterministic, it uniquely generates exactly one proof tree for each clausal representation of formulas, and, moreover, it uses some syntactic reductions of prefixes. TK is defined in the original methodology of tableau systems, without any external technique such as backtracking, backjumping, etc. Since all the necessary bookkeeping is built into the rules, the system is not only a basis for a validity algorithm, but is itself a decision procedure. We present also a deterministic tableau decision procedure which is an extension of TK and can be used for the global assumptions problem.