For a successful checkpointing strategy, we should place checkpoints so as to optimize fault-tolerance capability of real-time systems. This paper presents a novel scheme of checkpoint placement for real-time systems with periodic multi-tasks. Under the influence of transient faults, multi-tasks are scheduled by the Rate Monotonic (RM) algorithm. The optimal checkpoint intervals are derived to maximize the probability of task completion. In particular, this paper is concerned about the general case that the deadline of a task is longer than the period. Compared with the special condition that the deadline is equal to or less than the period, this general case causes a more complicate test procedure for schedulability of the RM algorithm with respect to a given set of checkpoint re-execution vectors. The probability of task completion is also derived in a more complex form. A case study is given to show the applicability of the proposed scheme.