The question requires matching the mathematical representations of different modulation schemes in LIST-I with their corresponding names in LIST-II.

Matching Modulation Schemes

We analyze each item in LIST-I:

• A. $[1+K_m(t)] A \sin(\omega_c t)$: This equation represents a carrier signal $A \sin(\omega_c t)$ whose amplitude is varied by a factor related to the message signal $K_m(t)$. Specifically, the term $[1+K_m(t)]$ modifies the amplitude. This is the standard form of Amplitude Modulation (AM). Thus, A matches with III.
• B. $K_m(t) A \sin(\omega_c t)$: Here, the amplitude of the carrier signal $A \sin(\omega_c t)$ is directly proportional to the message signal $K_m(t)$. This form, where the carrier component is suppressed, represents DSB-SC (Double Sideband Suppressed Carrier) modulation. Thus, B matches with IV.
• C. $A\sin[\omega_c t + K_m(t)]$: In this equation, the argument of the sine function represents the phase of the carrier wave. The term $K_m(t)$ is added directly to the carrier frequency term $\omega_c t$, indicating that the phase is modulated by the message signal. This is Phase Modulation (PM). Thus, C matches with I.
• D. $A\sin\!\left[\omega_c t + K \int_{-\infty}^{t} m(t)\, dt \right]$: This equation shows the phase of the carrier signal being varied by the integral of the message signal $m(t)$. A key characteristic of Frequency Modulation (FM) is that the instantaneous frequency is varied linearly with the message signal. The instantaneous phase is the integral of the instantaneous angular frequency. Therefore, modulating the phase by the integral of the message signal results in Frequency Modulation. Thus, D matches with II.

Correct Matching

Based on the analysis:

• A corresponds to III (Amplitude modulation)
• B corresponds to IV (DSB-SC modulation)
• C corresponds to I (Phase modulation)
• D corresponds to II (Frequency modulation)

The correct combination is A-III, B-IV, C-I, D-II.