The amplitude of `15sin(1000 pi t)` is modulated by `10sin(4 pi t)` signal.The amplitude modulated signal contains frequency (ies) of A `500Hz,` B. `2Hz,` C. `250Hz,` D. `498Hz` E. `502Hz,` D. Choose the correct answer from the options given below:

Options

A. B Only
B. `A` and `B` only `
C. `A,D` and `E` Only ``
D. A Only

Answer

To solve the problem of finding the frequencies present in the amplitude modulated signal, we will follow these steps:

### Step 1: Identify the carrier and modulating signals
The given signals are:
- Carrier signal: \( A_c(t) = 15 \sin(1000 \pi t) \)
- Modulating signal: \( A_m(t) = 10 \sin(4 \pi t) \)

### Step 2: Determine the frequencies of the signals
The frequency of the carrier signal (\( f_c \)) can be determined from its angular frequency:
\[ \omega_c = 1000 \pi \implies f_c = \frac{1000 \pi}{2 \pi} = 500 \text{ Hz} \]

The frequency of the modulating signal (\( f_m \)) can be determined similarly:
\[ \omega_m = 4 \pi \implies f_m = \frac{4 \pi}{2 \pi} = 2 \text{ Hz} \]

### Step 3: Apply the formula for frequencies in amplitude modulation
In amplitude modulation, the frequencies present in the modulated signal are:
\[ f_c + f_m \quad \text{and} \quad f_c - f_m \]
Calculating these frequencies:
1. \( f_c + f_m = 500 \text{ Hz} + 2 \text{ Hz} = 502 \text{ Hz} \)
2. \( f_c - f_m = 500 \text{ Hz} - 2 \text{ Hz} = 498 \text{ Hz} \)

### Step 4: List the frequencies present in the modulated signal
The frequencies present in the amplitude modulated signal are:
- \( 502 \text{ Hz} \)
- \( 498 \text{ Hz} \)

### Step 5: Analyze the options given
The options provided are:
- A. 500 Hz
- B. 2 Hz
- C. 250 Hz
- D. 498 Hz
- E. 502 Hz

From our calculations, the frequencies present in the modulated signal are \( 498 \text{ Hz} \) and \( 502 \text{ Hz} \).

### Conclusion
The correct answers from the options given are:
- D. 498 Hz
- E. 502 Hz