Choose the correct statement:

We start by recalling the basic definitions of the two most common analog modulation techniques used in communication systems, namely amplitude modulation (AM) and frequency modulation (FM).

In amplitude modulation, we take a high-frequency carrier wave whose general mathematical form can be written as

$$c(t)=A_c \sin\bigl(2\pi f_c t\bigr),$$

where $$A_c$$ represents the amplitude of the carrier, and $$f_c$$ denotes the carrier’s frequency. In AM, we superimpose an audio or base-band signal $$m(t)$$ on this carrier in such a way that the amplitude of the carrier becomes a function of the instantaneous value of the message signal, while the carrier’s frequency remains fixed. Symbolically, the modulated wave in AM can be expressed as

$$s_{\text{AM}}(t)=\bigl[A_c + k_a m(t)\bigr]\sin\bigl(2\pi f_c t\bigr),$$

where $$k_a$$ is the amplitude-sensitivity constant. We observe that the factor multiplied with $$\sin(2\pi f_c t)$$—namely $$A_c + k_a m(t)$$—changes directly with the amplitude of the audio signal $$m(t)$$. The carrier frequency $$f_c$$ does not change.

Next, in frequency modulation, we again start from the same carrier $$c(t)=A_c \sin(2\pi f_c t)$$, but here we keep the amplitude $$A_c$$ constant and allow the frequency term inside the sine function to vary with the message signal. The standard FM expression is

$$s_{\text{FM}}(t)=A_c \sin\Bigl(2\pi f_c t + k_f \int m(t)\,dt\Bigr),$$

where $$k_f$$ is the frequency-sensitivity constant. Thus, in FM, it is the frequency deviation that follows the instantaneous amplitude of $$m(t)$$, while the carrier amplitude stays unchanged.

With these well-established definitions in mind, we examine each option:

Option A states that in FM the amplitude of the carrier varies with the amplitude of the audio signal. This contradicts the FM definition, because FM keeps amplitude fixed and varies frequency. Therefore, Option A is incorrect.

Option B claims that in FM the amplitude of the carrier varies with the frequency of the audio signal. Again, FM does not touch the amplitude at all; it is the carrier frequency that is made to follow the amplitude (not the frequency) of the message. Hence Option B is also incorrect.

Option C states that in AM the amplitude of the carrier varies in proportion to the amplitude of the audio signal. As derived from the AM equation $$s_{\text{AM}}(t)=\bigl[A_c + k_a m(t)\bigr]\sin(2\pi f_c t)$$, this statement is exactly what AM does. Therefore, Option C is correct.

Option D says that in AM the frequency of the carrier varies with the amplitude of the audio signal. This would actually be a frequency-shift keyed idea which is not AM. Hence Option D is incorrect.

Only one statement aligns perfectly with the theoretical descriptions and mathematical formulations of AM and FM, and that is Option C.

Hence, the correct answer is Option C.