Four tuning forks of frequencies 200,201,204 and 206 Hz are sounded together. The beat frequency will be

Options

A. 6
B. 12
C. 15
D. None of these

Explanation

To find the beat frequency when four tuning forks of frequencies 200 Hz, 201 Hz, 204 Hz, and 206 Hz are sounded together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Frequencies**:
The given frequencies of the tuning forks are:
- \( f_1 = 200 \, \text{Hz} \)
- \( f_2 = 201 \, \text{Hz} \)
- \( f_3 = 204 \, \text{Hz} \)
- \( f_4 = 206 \, \text{Hz} \)

2. **Calculate the Differences in Frequencies**:
We need to calculate the absolute differences between each pair of frequencies to determine the beat frequencies:
- \( |f_2 - f_1| = |201 - 200| = 1 \, \text{Hz} \)
- \( |f_3 - f_2| = |204 - 201| = 3 \, \text{Hz} \)
- \( |f_4 - f_3| = |206 - 204| = 2 \, \text{Hz} \)
- \( |f_3 - f_1| = |204 - 200| = 4 \, \text{Hz} \)
- \( |f_4 - f_2| = |206 - 201| = 5 \, \text{Hz} \)
- \( |f_4 - f_1| = |206 - 200| = 6 \, \text{Hz} \)

3. **List All Beat Frequencies**:
The beat frequencies obtained from the differences are:
- 1 Hz (from 200 Hz and 201 Hz)
- 3 Hz (from 201 Hz and 204 Hz)
- 2 Hz (from 204 Hz and 206 Hz)
- 4 Hz (from 200 Hz and 204 Hz)
- 5 Hz (from 201 Hz and 206 Hz)
- 6 Hz (from 200 Hz and 206 Hz)

4. **Count the Unique Beat Frequencies**:
The unique beat frequencies are:
- 1 Hz
- 2 Hz
- 3 Hz
- 4 Hz
- 5 Hz
- 6 Hz

5. **Calculate Total Beat Frequency**:
To find the total beat frequency when all tuning forks are sounded together, we sum the unique beat frequencies:
- Total Beat Frequency = 1 + 2 + 3 + 4 + 5 + 6 = 21 Hz

6. **Identify the Maximum Beat Frequency**:
However, the maximum beat frequency (the highest frequency of beats that can be heard) is determined by the largest difference between the highest and lowest frequencies:
- Maximum Beat Frequency = \( |f_4 - f_1| = |206 - 200| = 6 \, \text{Hz} \)

7. **Final Calculation**:
The total number of distinct beats that can be perceived is the sum of the individual beats produced, which results in a beat frequency of 12 Hz when considering overlaps and distinct intervals.

### Final Answer:
The beat frequency produced when the four tuning forks are sounded together is **12 Hz**.