Periodic Motion & SHM Basics question from JEE Main 2026 (5 Apr, Shift 2)

Answer: (a)

1. A: $\sin^2\omega t = \frac{1 - \cos 2\omega t}{2}$, SHM about $\frac{1}{2}$ with $T = \frac{\pi}{\omega}$ $\to$ III.
2. B: $\sin^3(2\omega t)$ repeats with the period of $\sin 2\omega t$, $T = \frac{\pi}{\omega}$, but contains $\sin 6\omega t$ too, so not SHM $\to$ I.
3. C: $\sin\omega t + \cos(\pi\omega t)$: frequency ratio $1 : \pi$ is irrational, so no common period $\to$ IV.
4. D: $\cos\omega t + \cos 2\omega t$: common period $\frac{2\pi}{\omega}$, two frequencies so not SHM $\to$ II.
5. So the answer is (a) A-III, B-I, C-IV, D-II.