A source of unknown frequency gives 4 beats//s, when sounded with a source of known frequency 250 Hz. The second harmonic of the source of unknown frequency gives five beats per second, when sounded with a source of frequency 513 Hz .The unknown frequency is

Options

A. 254 Hz
B. 246 Hz
C. 240 Hz
D. 260 Hz

Explanation

To solve the problem, we need to determine the unknown frequency based on the information given about beats.

### Step 1: Understand the beats phenomenon
When two sound waves of different frequencies interfere, they produce a phenomenon called beats. The number of beats per second is equal to the absolute difference between the frequencies of the two sources.

### Step 2: Set up the equations for the first scenario
We know that the unknown frequency (let's denote it as \( f_u \)) gives 4 beats per second with a known frequency of 250 Hz. This can be expressed as:
\[ |f_u - 250| = 4 \]
This equation leads to two possible cases:
1. \( f_u - 250 = 4 \) → \( f_u = 254 \) Hz
2. \( 250 - f_u = 4 \) → \( f_u = 246 \) Hz

### Step 3: Set up the equations for the second scenario
Next, we consider the second harmonic of the unknown frequency, which is \( 2f_u \). This frequency gives 5 beats per second when sounded with a source of frequency 513 Hz. Thus, we have:
\[ |2f_u - 513| = 5 \]
This leads to two more cases:
1. \( 2f_u - 513 = 5 \) → \( 2f_u = 518 \) → \( f_u = 259 \) Hz
2. \( 513 - 2f_u = 5 \) → \( 2f_u = 508 \) → \( f_u = 254 \) Hz

### Step 4: Analyze the results
From the first scenario, we found two possible values for \( f_u \): 254 Hz and 246 Hz. From the second scenario, we found two possible values: 259 Hz and 254 Hz.

### Step 5: Determine the correct frequency
The only value that is consistent across both scenarios is \( f_u = 254 \) Hz.

### Conclusion
Thus, the unknown frequency is:
\[ \boxed{254 \text{ Hz}} \]