To solve the question regarding the type of metamorphosis shown by cockroaches, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding Metamorphosis**:
- Metamorphosis is the process by which an organism undergoes significant changes in its form and structure during its life cycle.
2. **Identifying the Stages of Cockroach Development**:
- The life cycle of a cockroach consists of three main stages:
- **Egg**: The initial stage where cockroach eggs are laid.
- **Nymph**: The immature form that resembles a smaller version of the adult but lacks wings and reproductive organs.
- **Adult**: The final stage where the cockroach is fully developed with wings and reproductive capabilities.
3. **Classifying the Type of Metamorphosis**:
- Metamorphosis can be classified as:
- **Complete Metamorphosis**: Involves four distinct stages: egg, larva, pupa, and adult (e.g., butterflies).
- **Incomplete Metamorphosis**: Involves three stages: egg, nymph, and adult (e.g., cockroaches, grasshoppers).
4. **Analyzing the Options**:
- The options provided are:
- A) Complete with egg and adult stages
- B) Complete with egg, nymph, and adult stages
- C) Incomplete with egg, nymph, and adult stages
- D) Incomplete with egg and adult stages
- Since we established that cockroaches undergo incomplete metamorphosis, we can eliminate options A and B as they suggest complete metamorphosis.
5. **Selecting the Correct Answer**:
- The remaining options are C and D. Since cockroaches do have nymph stages, option D is incorrect.
- Therefore, the correct answer is **C) Incomplete with egg, nymph, and adult stages**.
### Final Answer:
The metamorphosis shown by cockroaches is **C) Incomplete with egg, nymph, and adult stages**.
