In card-based cryptography, a commitment to a Boolean value is usually represented by two face-down cards of different colors or numbers, whose order specifies the one-bit value (namely, 0 or 1). One of the most important primitives in card-based cryptography is a "copy protocol,"which is supposed to make two identical copies of a given commitment. In the literature, there are several copy protocols, which can be categorized by kinds of shuffles they use; this paper focuses on those using only the so-called random cut, which is the simplest shuffle, and we propose two copy protocols that are more efficient than the existing ones. Specifically, we first work on a standard deck of cards and design a six-card copy protocol using three random cuts (on average). Since the previous protocol needs 5.5 random cuts, our protocol improves upon it. Next, we shift our attention to the case of a two-colored deck of cards, and construct a six-card copy protocol using three random cuts (on average). Because the previous protocol requires eight cards, our protocol uses two cards fewer than the previous one (although it uses one more shuffle). In addition, going back to the standard-deck setting, we provide a four-card XOR protocol using only one random cut for the first time.