Two identical massless springs A and B consist spring constant `k_(A)` and `k_(B)` respectively. Then :

Options

A. it they are compressed by same force, work done on A is more expanded when `k_(A) gt k_(B)`
B. if they are compressed by same amount, work done on A is more expanded when `k_(A)lt k_(B)`
C. if they are compressed by same amount, work done on A is more expanded when `k_(A) gt k_(B)`
D. both (1) and (2) are correct

Answer

To solve the problem, we need to analyze the relationship between the spring constants \( k_A \) and \( k_B \) of two identical massless springs A and B, and how the work done on these springs varies based on the conditions provided in the options.

### Step-by-Step Solution:

1. **Understanding the Potential Energy in Springs**:
The potential energy \( U \) stored in a spring when it is compressed or stretched is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

2. **Analyzing Option A**:
- The option states that the springs are compressed by the same force.
- The work done \( W \) on a spring is related to the force applied and the displacement.
- Since \( W = F \cdot x \) and \( F = kx \), we can express the work done in terms of the spring constant:
\[ W = \frac{F^2}{2k} \]
- If \( k_A > k_B \), then for the same force, the work done on spring A would be less than that on spring B. Thus, this option is incorrect.

3. **Analyzing Option B**:
- This option states that the springs are compressed by the same amount \( x \).
- Using the potential energy formula, if \( k_A > k_B \), then:
\[ U_A = \frac{1}{2} k_A x^2 \quad \text{and} \quad U_B = \frac{1}{2} k_B x^2 \]
- Since \( k_A > k_B \), it follows that \( U_A > U_B \). Hence, the work done on A is greater than on B, making this option incorrect.

4. **Analyzing Option C**:
- This option also states that the springs are compressed by the same amount \( x \).
- Again using the potential energy formula, if \( k_A > k_B \):
\[ U_A = \frac{1}{2} k_A x^2 \quad \text{and} \quad U_B = \frac{1}{2} k_B x^2 \]
- Since \( k_A > k_B \), we have \( U_A > U_B \). Therefore, the work done on spring A is indeed greater than on spring B, confirming that this option is correct.

5. **Analyzing Option D**:
- Without specific details provided in the transcript, we can infer that this option is likely incorrect based on the analysis of the previous options.

### Conclusion:
The correct answer is **Option C**, as it correctly states that if the springs are compressed by the same amount and \( k_A > k_B \), then the work done on spring A is greater than the work done on spring B.