Match List-I with List-II.

List-I	List-II
A. \(\sin^2 \omega t\)	I. Periodic with time period \(T = \dfrac{\pi}{\omega}\) but not simple harmonic motion (SHM)
B. \(\sin^3(2\omega t)\)	II. Periodic with time period \(T = \dfrac{2\pi}{\omega}\) but Not SHM
C. \(\sin(\omega t) + \cos(\pi \omega t)\)	III. Periodic with time period \(T = \dfrac{\pi}{\omega}\) and SHM
D. \(\cos\omega t + \cos 2\omega t\)	IV. Non-periodic

Choose the correct answer from the options given below :

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