To solve the question regarding the correct statements about a cockroach, let's analyze each statement step by step.
### Step 1: Analyze the first statement
**Statement:** The sclerites join to each other by a thick, hard and non-flexible arthroidal membrane.
**Analysis:** This statement is incorrect. The sclerites of a cockroach are joined by a thin, flexible arthroidal membrane, not a thick, hard, and non-flexible one.
**Hint:** Remember that the flexibility of the arthroidal membrane allows for movement between sclerites.
### Step 2: Analyze the second statement
**Statement:** Each thoracic segment bears a pair of flying wings.
**Analysis:** This statement is also incorrect. Each thoracic segment of a cockroach bears a pair of legs. The wings are present only on the mesothoracic segment.
**Hint:** Recall the anatomy of the cockroach; only the second thoracic segment has wings.
### Step 3: Analyze the third statement
**Statement:** In the cockroach, the fat body, nephrocytes, and uricose glands help in excretion.
**Analysis:** This statement is correct. The fat body, nephrocytes, and uricose glands are indeed involved in the excretory process in cockroaches, helping in the removal of waste.
**Hint:** Consider the role of these structures in the metabolic waste removal process.
### Step 4: Analyze the fourth statement
**Statement:** A characteristic mushroom gland is present in the 4th to 6th abdominal segment.
**Analysis:** This statement is incorrect. The mushroom gland is actually located in the 6th to 7th abdominal segment, not the 4th to 6th.
**Hint:** Pay attention to the specific segments where various glands are located in the cockroach.
### Conclusion
Based on the analysis of each statement, the only correct statement is:
- In the cockroach, the fat body, nephrocytes, and uricose glands help in excretion.
### Summary of Correct Statements
1. The first statement is incorrect.
2. The second statement is incorrect.
3. The third statement is correct.
4. The fourth statement is incorrect.
### Final Answer
The correct statement about a cockroach is: **In the cockroach, the fat body, nephrocytes, and uricose glands help in excretion.**
