Two tuning forks x and y produce tones of frequencies 256 Hz and 262 Hz respectively, An unknown tone sounded with x produces, beats. When it is sounded with y the number of beats produced is doubled. The unknown frequency is

Options

A. 254 Hz
B. 258 Hz
C. 264 Hz
D. 259 Hz

Answer

To find the unknown frequency, we will follow these steps:

### Step 1: Define the known frequencies
Let:
- Frequency of tuning fork X (f_X) = 256 Hz
- Frequency of tuning fork Y (f_Y) = 262 Hz
- Unknown frequency (f) = F

### Step 2: Set up the equations for beats
When the unknown frequency F is sounded with tuning fork X, the number of beats produced is N:
\[ |f_X - F| = N \]
This can be expressed as:
\[ 256 - F = N \quad \text{(1)} \]
or
\[ F - 256 = N \quad \text{(2)} \]

When the unknown frequency F is sounded with tuning fork Y, the number of beats produced is doubled (2N):
\[ |f_Y - F| = 2N \]
This can be expressed as:
\[ 262 - F = 2N \quad \text{(3)} \]
or
\[ F - 262 = 2N \quad \text{(4)} \]

### Step 3: Analyze the equations
We will consider two cases based on equations (1) and (3):
1. From equation (1): \( F = 256 - N \)
2. Substitute this into equation (3):
\[ 262 - (256 - N) = 2N \]
\[ 262 - 256 + N = 2N \]
\[ 6 + N = 2N \]
\[ N = 6 \]

### Step 4: Substitute N back to find F
Now substitute N back into equation (1):
\[ 256 - F = 6 \]
\[ F = 256 - 6 \]
\[ F = 250 \text{ Hz} \]

### Conclusion
The unknown frequency is:
\[ \boxed{250 \text{ Hz}} \]

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