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(C, B) For $f(x) = \sin(\pi \cos x)$,$f(x) = 0 \implies \pi \cos x = n\pi \implies \cos x = n$. Since $n \in \mathbb{Z}$ and $|\cos x| \le 1$,$n \in \

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$3, 2$

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(C, B) For $f(x) = \sin(\pi \cos x)$,$f(x) = 0 \implies \pi \cos x = n\pi \implies \cos x = n$. Since $n \in \mathbb{Z}$ and $|\cos x| \le 1$,$n \in \{-1, 0, 1\}$. Thus $x = n\pi$ or $x = n\pi \pm \frac{\pi}{2}$,which simplifies to $x = \frac{k\pi}{2}$ for $k \in \mathbb{N}$. $X = \{\frac{k\pi}{2} : k \in \mathbb{N}\}$. This is an arithmetic progression with common difference $\frac{\pi}{2}$. Thus $(I) 	o (Q)$.For $f'(x) = \cos(\pi \cos x) \cdot (-\pi \sin x) = 0$. Either $\sin x = 0 \implies x = n\pi$ or $\cos(\pi \cos x) = 0 \implies \pi \cos x = (2n+1)\frac{\pi}{2} \implies \cos x = \pm \frac{1}{2}$. $Y = \{n\pi, n\pi \pm \frac{\pi}{3}\}$. This is not an $AP$. $Y$ contains $\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\}$. Thus $(II) 	o (R), (T)$.For $g(x) = \cos(2\pi \sin x) = 0 \implies 2\pi \sin x = (2n+1)\frac{\pi}{2} \implies \sin x = \pm \frac{1}{4}, \pm \frac{3}{4}$. This is not an $AP$. Thus $(III) 	o (R)$.For $g'(x) = -\sin(2\pi \sin x) \cdot (2\pi \cos x) = 0$. Either $\cos x = 0 \implies x = (2n+1)\frac{\pi}{2}$ or $\sin(2\pi \sin x) = 0 \implies 2\pi \sin x = n\pi \implies \sin x = \frac{n}{2}$. Thus $\sin x \in \{0, \pm \frac{1}{2}, \pm 1\}$. $W$ contains $\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\}$. Thus $(IV) 	o (S)$.Matching: $(1)$ $(II), (R), (T)$ is not an option,but $(II), (R), (T)$ is correct. Re-evaluating options: $(1)$ $(II), (R), (S)$ is false. $(2)$ $(I), (P), (R)$ is false. $(3)$ $(II), (Q), (T)$ is false. $(4)$ $(I), (Q), (U)$ is correct as $X$ is an $AP$ and $U$ is a subset of $X$. For $(2)$,$(III), (R), (U)$ is correct as $Z$ is not an $AP$ and $U$ is a subset of $Z$ (since $\sin(\pi/6)=1/2$ and $\sin(3\pi/4)=1/\sqrt{2}$,wait,$Z$ requires $\sin x = \pm 1/4, \pm 3/4$. Re-checking $Z$: $g(x)=0 \implies \sin x = \pm 1/4, \pm 3/4$. $U$ is not a subset of $Z$. Correcting: $(IV), (P), (R), (S)$ is correct for $(2)$.

$List-I$	$List-II$
$(I) X$	$(P) \supseteq \{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\}$
$(II) Y$	$(Q) \text{ an arithmetic progression}$
$(III) Z$	$(R) \text{ NOT an arithmetic progression}$
$(IV) W$	$(S) \supseteq \{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\}$
	$(T) \supseteq \{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\}$
	$(U) \supseteq \{\frac{\pi}{6}, \frac{3\pi}{4}\}$

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