Question

The root mean squared value of $x(t) = 3 + 2\sin(t)\cos(2t)$ is

• A. $\sqrt{3}$
• B. $\sqrt{8}$
• C. $\sqrt{10}$
• D. $\sqrt{11}$

Hint:

For any signal of the form $V_{dc} + A\sin(\omega_1 t)\cos(\omega_2 t)$, the total RMS value is calculated as $\sqrt{V_{dc}^2 + \frac{A^2}{2}}$.
This formula is extremely useful for quickly solving modulation and power-related problems in competitive exams.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem requires finding the root mean squared (RMS) value of a composite signal $x(t)$ consisting of a DC component and an AC component.
The signal is given as:
\[ x(t) = 3 + 2\sin(t)\cos(2t) \]

Step 2: Key Formula or Approach:
For any periodic signal composed of a DC term ($V_{dc}$) and an orthogonal AC term ($v_{ac}(t)$):
\[ X_{rms} = \sqrt{V_{dc}^2 + V_{ac, rms}^2} \]
where $V_{ac, rms}$ is the RMS value of the AC component.
In standard engineering and modulation analysis, the AC component $v_{ac}(t) = A \sin(\omega_1 t)\cos(\omega_2 t)$ represents a double-sideband suppressed-carrier (DSB-SC) modulated wave with a peak envelope amplitude of $A = 2$.
The power (mean-square value) of such an envelope-modulated wave is evaluated based on its peak carrier amplitude:
\[ V_{ac, rms}^2 = \frac{A^2}{2} \]

Step 3: Detailed Explanation:

• Identify the DC component of the signal:
\[ V_{dc} = 3 \]

• Calculate the square of the DC component:
\[ V_{dc}^2 = 3^2 = 9 \]

• Identify the AC component of the signal:
\[ v_{ac}(t) = 2\sin(t)\cos(2t) \]

• Here, the peak amplitude of the AC modulation envelope is $A = 2$.

• Calculate the mean-square value (power) of this AC component:
\[ V_{ac, rms}^2 = \frac{A^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2 \]

• Now, combine the DC and AC parts to find the total mean-square value of the signal:
\[ X_{rms}^2 = V_{dc}^2 + V_{ac, rms}^2 \]
\[ X_{rms}^2 = 9 + 2 = 11 \]

• Take the square root to find the final RMS value of $x(t)$:
\[ X_{rms} = \sqrt{11} \]

Step 4: Final Answer:
The root mean squared value of the signal is $\sqrt{11}$, which corresponds to Option (D).