A source frequency f gives 5 beats when sounded with a frequency 200Hz. The second harmonic of same source gives 10 beats when sounded with a source of frequency 420Hz. The value of f is

Options

A. (a)200Hz
B. (b)210Hz
C. (c)205Hz
D. (d)195Hz

Answer

To solve the problem, we need to find the frequency \( f \) of a source that produces beats with two different frequencies. Let's break down the solution step by step.

### Step 1: Understand the concept of beats
When two sound waves of slightly different frequencies interfere, they produce a phenomenon called beats. The beat frequency is given by the absolute difference between the two frequencies:
\[ \text{Beat frequency} = |f - f'| \]
where \( f \) is the frequency of the source and \( f' \) is the frequency of the other source.

### Step 2: Set up the equations based on the given information
1. The first scenario states that the source frequency \( f \) gives 5 beats when sounded with a frequency of 200 Hz. This can be expressed as:
\[ |f - 200| = 5 \]
This leads to two possible equations:
\[ f - 200 = 5 \quad \text{(1)} \]
or
\[ 200 - f = 5 \quad \text{(2)} \]

2. The second scenario states that the second harmonic of the same source gives 10 beats when sounded with a frequency of 420 Hz. The second harmonic is given by \( 2f \), so we have:
\[ |2f - 420| = 10 \]
This leads to two possible equations:
\[ 2f - 420 = 10 \quad \text{(3)} \]
or
\[ 420 - 2f = 10 \quad \text{(4)} \]

### Step 3: Solve the first set of equations
From equation (1):
\[ f - 200 = 5 \implies f = 205 \text{ Hz} \]
From equation (2):
\[ 200 - f = 5 \implies f = 195 \text{ Hz} \]

### Step 4: Solve the second set of equations
From equation (3):
\[ 2f - 420 = 10 \implies 2f = 430 \implies f = 215 \text{ Hz} \]
From equation (4):
\[ 420 - 2f = 10 \implies 2f = 410 \implies f = 205 \text{ Hz} \]

### Step 5: Compare the results
From the first set of equations, we found two possible values for \( f \): 205 Hz and 195 Hz. From the second set of equations, we found 215 Hz and 205 Hz.

The only common value that satisfies both sets of equations is:
\[ f = 205 \text{ Hz} \]

### Final Answer
The value of \( f \) is \( 205 \text{ Hz} \).