We show that, for every integer k≥ 4 , if M is a k-connected matroid and C is a circuit of M such that for every e∈ C, M\ e is not k-connected, then C meets a cocircuit of size at most 2 k- 3 ; furthermore, if M is binary and k≥ 5 , then C meets a cocircuit of size at most 2 k- 4. It follows from our results and a result of Reid et al [5] that every minimally k-connected matroid has a cocircuit of size at most 2 k- 3 , and every minimally k-connected binary matroid has a cocircuit of size at most 2 k- 4.