Periplaneta is a genus of cockroaches, commonly known as the American cockroach. Cockroaches undergo incomplete metamorphosis, which means they do not go through a complete transformation from egg to adult like insects that undergo complete metamorphosis. Instead, they undergo gradual metamorphosis.

What is Metamorphosis?
Metamorphosis is the process of transformation that insects go through as they develop from egg to adult. It involves distinct stages of growth and development.

Complete Metamorphosis:
In insects that undergo complete metamorphosis, there are four distinct stages:
1. Egg: The insect begins its life cycle as an egg.
2. Larva: After hatching from the egg, the insect goes through a larval stage. The larva looks significantly different from the adult and often has different feeding habits.
3. Pupa: The larva then enters a pupal stage. During this stage, the insect undergoes a complete reorganization of its body structure and transforms into the adult form.
4. Adult: Finally, the insect emerges from the pupa as a fully developed adult.

Incomplete Metamorphosis:
Insects that undergo incomplete metamorphosis have three stages:
1. Egg: The insect starts its life cycle as an egg.
2. Nymph: After hatching from the egg, the insect becomes a nymph. Nymphs resemble the adult form, but lack fully developed wings and reproductive organs.
3. Adult: The nymph undergoes a series of molts, shedding its exoskeleton and growing larger with each molt. Eventually, it reaches adulthood.

Gradual Metamorphosis:
Gradual metamorphosis is similar to incomplete metamorphosis, but the immature stages, called nymphs, resemble the adult form more closely. The nymphs have similar body structures and habits to the adults, but are smaller in size and lack fully developed wings and reproductive organs. As the nymphs grow, they molt and continue to develop until they reach the adult stage.

Periplaneta and Metamorphosis:
Periplaneta, or American cockroach, undergoes incomplete metamorphosis. After hatching from the egg, the nymphs closely resemble the adult cockroaches in terms of body structure and habits. They continue to molt and grow until they reach adulthood. Therefore, the correct answer is option 'D' - No metamorphosis.

Periplaneta is an American cockroach and we know cockroaches shows incomplete metamorphosis . so according to me option b is correct