A): An open organ pipe of certain length have the same fundamental frequency as closed organ pipe of half the length.
(R): In the case of open organ pipe, at both the ends antinodes are formed, while in the closed organ pipe at one end antinode and at the other end node is formed.

Options

A. Both 'A' and 'R' are true and 'R' is the correct, explanation of 'A'.
B. Both 'A' and 'R' are true and 'R' is not the correct explanation of 'A'
C. A' is true and 'R' is false
D. Both 'A' and 'R' are false

Answer

To solve the question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step-by-Step Solution:

1. **Understanding the Assertion (A)**:
- The assertion states that an open organ pipe of a certain length has the same fundamental frequency as a closed organ pipe of half that length.
- Let's denote the length of the open organ pipe as \( L \) and the length of the closed organ pipe as \( L/2 \).

2. **Fundamental Frequency of Open Organ Pipe**:
- The formula for the fundamental frequency (\( f \)) of an open organ pipe is given by:
\[ f_{\text{open}} = \frac{V}{2L} \]
- Here, \( V \) is the speed of sound in air.

3. **Fundamental Frequency of Closed Organ Pipe**:
- The formula for the fundamental frequency of a closed organ pipe is:
\[ f_{\text{closed}} = \frac{V}{4L_{\text{closed}}} \]
- Since the length of the closed organ pipe is \( L/2 \), we substitute this into the formula:
\[ f_{\text{closed}} = \frac{V}{4 \cdot (L/2)} = \frac{V}{2L} \]

4. **Comparison of Frequencies**:
- From the calculations, we find:
\[ f_{\text{open}} = \frac{V}{2L} \quad \text{and} \quad f_{\text{closed}} = \frac{V}{2L} \]
- This shows that the fundamental frequencies of both pipes are equal, confirming the assertion.

5. **Understanding the Reason (R)**:
- The reason states that in the case of an open organ pipe, antinodes are formed at both ends, while in a closed organ pipe, one end is an antinode and the other end is a node.
- This statement is true and describes the boundary conditions for both types of pipes accurately.

6. **Conclusion**:
- Both the assertion (A) and the reason (R) are correct.
- However, the reason does not explain why the frequencies are the same, thus the correct answer is that both statements are true, but the reason does not adequately support the assertion.

### Final Answer:
Both assertion and reason are correct, but the reason does not explain the assertion.