The correct mathematical expression for amplitude modulated wave is :

The correct answer is Option (3) → \( (A_c + A_m\sin\omega_m t)\sin\omega_c t \)

In amplitude modulation the amplitude of the high-frequency carrier wave \( A_c\sin\omega_c t \) is varied in step with the message signal \( A_m\sin\omega_m t \). The message is added to the carrier's amplitude, and the carrier's frequency stays the same.

    \( \displaystyle c_m(t) = (A_c + A_m\sin\omega_m t)\sin\omega_c t \)

Option (1) \( A_c\sin\omega_c t \) — Incorrect. This is the unmodulated carrier wave alone; its amplitude \( A_c \) does not vary, so it carries no message.

Option (2) \( A_m\sin\omega_m t \) — Incorrect. This is the message (modulating) signal alone, a low-frequency wave that has not been put on a carrier.

Option (3) \( (A_c + A_m\sin\omega_m t)\sin\omega_c t \) — Correct. The amplitude of the carrier, \( A_c + A_m\sin\omega_m t \), varies with the message signal while the carrier frequency \( \omega_c \) stays fixed.

Option (4) \( A_c\sin\omega_c t + A_m\sin\omega_m t \) — Incorrect. This is only the sum (superposition) of the two waves. Each keeps its own amplitude, so the carrier's amplitude is not modulated.

Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.

NCERT (Pre-rationalised edition, 2022-23):
Amplitude Modulation – “… Let c(t) = A_c sin ω_ct represent carrier wave and m(t) = A_m sin ω_mt represent the message or the modulating signal where ω_m = 2πf_m is the angular frequency of the message signal. The modulated signal c_m(t) can be written as

    \( \displaystyle c_m(t) = (A_c + A_m \sin\omega_m t)\sin\omega_c t = A_c\left(1 + \frac{A_m}{A_c}\sin\omega_m t\right)\sin\omega_c t \qquad (15.3) \)”