$\text{Consider a real, narrowband signal}\mathrm{ x}\left(t\right)=A\left(t\right)\cos [2\pi f_{c}t+\theta (t\left)\right]\text{where the maximum frequency components}$

$\text{of}\mathrm{ A}\left(t\right)\text{ and }\mathrm{\theta }\left(t\right)\text{are}{\mathrm{ f}}_M\text{and}{\mathrm{ f}}_c(=1000f_M)\text{, respectively.}$ $\text{Which of the following statements is/are correct?}$

The correct statement is: x(t) represents an amplitude modulated signal for suitable choices of A(t) and θ(t).

Let us analyze the general representation of a bandpass or modulated signal. A real, narrowband signal $x\left(t\right)$ is expressed in its carrier-modulated form as:
$x\left(t\right)=A\left(t\right)\cos [2\pi f_{c}t+\theta (t\left)\right]$
where:
1. $A\left(t\right)$ represents the time-varying envelope (amplitude variation).
2. $\theta \left(t\right)$ represents the time-varying phase deviation.
3. $f_c$ is the carrier frequency.

By selecting suitable functions for $A\left(t\right)$ and $\theta \left(t\right)$, this general equation can describe various analog and digital modulation schemes:
- If we choose $A\left(t\right)=A_c[1+m(t\left)\right]$ (where $m\left(t\right)$ is the message signal) and a constant phase $\theta \left(t\right)={\theta }_0$, the signal $x\left(t\right)$ directly simplifies to a conventional Amplitude Modulated (AM) signal.
- Since such suitable choices of $A\left(t\right)$ and $\theta \left(t\right)$ are mathematically valid and physically realizable under the given narrowband constraints, the statement that $x\left(t\right)$ represents an amplitude modulated signal is correct.