List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in $\mathrm{m} {\mathrm{s}}^{-1}$) at time $t=\frac{\pi }{3} \mathrm{s}$.

	List-I		List-II
(I)	$A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{m}$ with uniform angular speed $\omega =1 \mathrm{rad} {\mathrm{s}}^{-1}$. The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta =0$ and $\theta =\frac{\pi }{2}$ respectively.	(P)	$\frac{\sqrt{3}+1}{2}$
(II)	Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{s}$ respectively, with the same speed $v=\frac{5\pi }{\sqrt{2}} \mathrm{m} {\mathrm{s}}^{-1}$ and at $45°$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide, $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$.	(Q)	$\frac{\left(\sqrt{3}-1\right)}{\sqrt{2}}$
(III)	Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_A=x_0\sin \frac{t}{t_0}$ and $x_B=x_0\sin \left(\frac{t}{t_0}+\frac{\pi }{2}\right)$ respectively, starting from $t=0$. Take $x_0=1 \mathrm{m},t_0=1 \mathrm{s}$.	(R)	$\sqrt{10}$
(IV)	Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{m}$ on the $xy$ plane, with constant angular speed $\omega =1 \mathrm{rad} {\mathrm{s}}^{-1}$. Particle $B$ is moving up at a constant speed $3 \mathrm{m} {\mathrm{s}}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S)	$\sqrt{2}$
		(T)	$\sqrt{25{\pi }^2+1}$

Which one of the following options is correct?

1. A $\mathrm{I}\to \mathrm{R},\mathrm{II}\to \mathrm{T},\mathrm{III}\to \mathrm{P},\mathrm{IV}\to \mathrm{S}$
2. B $\mathrm{I}\to \mathrm{S},\mathrm{II}\to \mathrm{P},\mathrm{III}\to \mathrm{Q},\mathrm{IV}\to \mathrm{R}$
3. C $\mathrm{I}\to \mathrm{S},\mathrm{II}\to \mathrm{T},\mathrm{III}\to \mathrm{P},\mathrm{IV}\to \mathrm{R}$
4. D $\mathrm{I}\to \mathrm{T},\mathrm{II}\to \mathrm{P},\mathrm{III}\to \mathrm{R},\mathrm{IV}\to \mathrm{S}$