Which of the following are correct?

Options

A. The product of bulk' modulus of elasticity and compressibility is `1`
B. A rope `1 cm` in diameter breaks if the tension in it exceeds `500 N`. The maximum tension that may be given to a similar rope of diameter `2 cm` is `2000 N`.
C. Both (a) and (b) are correct.
D. Neither (a) nor (b) is correct.

Explanation

To solve the question, we need to evaluate the correctness of the given options based on the principles of physics related to elasticity and stress.

### Step-by-Step Solution:

**Step 1: Evaluate Option A**
- **Statement**: The product of bulk modulus of elasticity and compressibility is 1.
- **Explanation**:
- Bulk modulus (K) is defined as the measure of a substance's resistance to uniform compression.
- Compressibility (β) is the measure of the relative volume change of a material in response to a pressure change, and it is the inverse of bulk modulus.
- Mathematically, we have:
\[ \beta = \frac{1}{K} \]
- Therefore, the product of bulk modulus and compressibility is:
\[ K \cdot \beta = K \cdot \frac{1}{K} = 1 \]
- **Conclusion**: Option A is correct.

**Step 2: Evaluate Option B**
- **Statement**: A rope of 1 cm in diameter breaks if the tension in it exceeds 500 Newton. The maximum tension that may be given to a similar rope of diameter 2 cm is 2000 N.
- **Explanation**:
- The maximum stress (σ_max) in the rope can be calculated using the formula:
\[ \sigma_{\text{max}} = \frac{F}{A} \]
where \( F \) is the force (tension) and \( A \) is the cross-sectional area of the rope.
- For a rope of diameter 1 cm:
\[ A_1 = \frac{\pi}{4} d^2 = \frac{\pi}{4} (1 \text{ cm})^2 = \frac{\pi}{4} \text{ cm}^2 \]
\[ \sigma_{\text{max}} = \frac{500 \text{ N}}{\frac{\pi}{4} \text{ cm}^2} = \frac{2000}{\pi} \text{ N/cm}^2 \]
- For a rope of diameter 2 cm:
\[ A_2 = \frac{\pi}{4} (2 \text{ cm})^2 = \frac{\pi}{4} \cdot 4 \text{ cm}^2 = \pi \text{ cm}^2 \]
- Let the maximum tension for the 2 cm rope be \( x \):
\[ \sigma_{\text{max}} = \frac{x}{\pi \text{ cm}^2} \]
- Equating the maximum stress from both cases:
\[ \frac{2000}{\pi} = \frac{x}{\pi} \]
- Cancelling \( \pi \) from both sides, we find:
\[ x = 2000 \text{ N} \]
- **Conclusion**: Option B is correct.

**Step 3: Evaluate Option C**
- **Statement**: Both A and B are correct.
- **Conclusion**: Since both options A and B have been verified to be correct, option C is also correct.

### Final Answer:
- **Correct Options**: A, B, and C are all correct.