Match List-I with List-II. List-I List-II A. ^2 ω t I. Periodic...

For A: $y = \sin^2 \omega t = \dfrac{1 - \cos 2\omega t}{2} = \dfrac{1}{2} - \dfrac{1}{2} \cos 2\omega t$

This represents simple harmonic motion (SHM) about the mean position $y = \dfrac{1}{2}$ with an angular frequency of $2\omega$. The time period is $T = \dfrac{2\pi}{2\omega} = \dfrac{\pi}{\omega}$. Thus, A matches III.

For B: $y = \sin^3(2\omega t) = \dfrac{3\sin 2\omega t - \sin 6\omega t}{4}$

This is a superposition of two SHMs with angular frequencies $2\omega$ and $6\omega$. It is periodic but not SHM. The time period is the LCM of $T_1 = \dfrac{2\pi}{2\omega} = \dfrac{\pi}{\omega}$ and $T_2 = \dfrac{2\pi}{6\omega} = \dfrac{\pi}{3\omega}$, which is $T = \dfrac{\pi}{\omega}$. Thus, B matches I.

For C: $y = \sin(\omega t) + \cos(\pi \omega t)$

The angular frequencies are $\omega_1 = \omega$ and $\omega_2 = \pi \omega$. The ratio $\dfrac{\omega_1}{\omega_2} = \dfrac{1}{\pi}$ is an irrational number. Therefore, the function is non-periodic. Thus, C matches IV.

For D: $y = \cos \omega t + \cos 2\omega t$

This is a superposition of two SHMs with angular frequencies $\omega$ and $2\omega$. It is periodic but not SHM. The time period is the LCM of $T_1 = \dfrac{2\pi}{\omega}$ and $T_2 = \dfrac{2\pi}{2\omega} = \dfrac{\pi}{\omega}$, which is $T = \dfrac{2\pi}{\omega}$. Thus, D matches II.

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

Answer: A-III, B-I, C-IV, D-II