Let $f(x) = \sin(\pi \cos x)$ and $g(x) = \cos(2\pi \sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order. $X = \{x : f(x) = 0\}$, $Y = \{x : f'(x) = 0\}$, $Z = \{x : g(x) = 0\}$, $W = \{x : g'(x) = 0\}$. Column-I contains the sets $X$, $Y$, $Z$ and $W$. Column-II contains some information regarding these sets. Column-I: (I) $X$, (II) $Y$, (III) $Z$, (IV) $W$. Column-II: (p) $\supseteq \{\pi/2, 3\pi/2, 4\pi, 7\pi\}$; (q) an arithmetic progression; (r) NOT an arithmetic progression; (s) $\supseteq \{\pi/6, 7\pi/6, 13\pi/6\}$; (t) $\supseteq \{\pi/3, 2\pi/3, \pi\}$; (u) $\supseteq \{\pi/6, 3\pi/4\}$.

$\text{Given: } f(x) = \sin(\pi \cos x), \quad g(x) = \cos(2\pi \sin x)$

$X = \{x : f(x) = 0\}, \qquad Y = \{x : f'(x) = 0\}$

$Z = \{x : g(x) = 0\}, \qquad W = \{x : g'(x) = 0\}$

$\text{To find: the matching of } X, Y, Z, W$

$\text{with the correct statements from Column II } (p)\text{--}(u)$

$1.\ \text{find zeros of } f,\ f',\ g,\ g'$

$2.\ \text{write each set and check AP}$

$3.\ \text{check membership and match options}$

$f(x)=0 \Rightarrow \sin(\pi\cos x)=0 \Rightarrow \pi\cos x = n\pi$

$\Rightarrow \cos x = n,\ n\in\mathbb{Z};\quad \cos x\in[-1,1]\Rightarrow n=-1,0,1$

$X = \left\{x:\cos x\in\{-1,0,1\}\right\} = \left\{\frac{m\pi}{2}: m\ge 1\right\}$

$X = \left\{\frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi, \frac{5\pi}{2}, \ldots\right\}$

$\text{successive gaps} = \frac{\pi}{2} \Rightarrow X \text{ is an AP} \Rightarrow (q)$

$\frac{\pi}{2},\ \frac{3\pi}{2},\ 4\pi,\ 7\pi \in X \Rightarrow X \supseteq (p)$

$f'(x) = -\pi\sin x\cos(\pi\cos x)$

$f'(x)=0 \Rightarrow \sin x = 0 \ \text{or}\ \cos(\pi\cos x)=0$

$\sin x=0 \Rightarrow x = n\pi$

$\cos(\pi\cos x)=0 \Rightarrow \pi\cos x = \frac{\pi}{2}+n\pi$

$\Rightarrow \cos x = \frac{1}{2}+n \Rightarrow \cos x = \pm\frac{1}{2}$

$\Rightarrow x = \pm\frac{\pi}{3}+2k\pi,\ \pm\frac{2\pi}{3}+2k\pi$

$Y = \left\{\frac{m\pi}{3}: m\ge1\right\} = \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}, \ldots\right\}$

$\text{common difference} = \frac{\pi}{3} \Rightarrow Y \text{ is an AP} \Rightarrow (q)$

$\frac{\pi}{3}, \frac{2\pi}{3}, \pi \in Y \Rightarrow Y \supseteq (t)$

$g(x)=0 \Rightarrow \cos(2\pi\sin x)=0 \Rightarrow 2\pi\sin x = \frac{\pi}{2}+n\pi$

$\Rightarrow \sin x = \frac{1}{4}+\frac{n}{2} = \pm\frac{1}{4},\ \pm\frac{3}{4}$

$Z = \left\{x:\sin x\in\left\{\pm\frac{1}{4},\pm\frac{3}{4}\right\}\right\}$

$\text{gaps between consecutive } Z\text{-elements are not equal}$

$\Rightarrow Z \text{ is NOT an AP} \Rightarrow (r)$

$\sin\frac{\pi}{2}=1 \Rightarrow \frac{\pi}{2}\notin Z$

$\sin\frac{\pi}{6}=\frac{1}{2} \Rightarrow \frac{\pi}{6}\notin Z$

$Z \text{ matches only } (r)$

$g'(x) = -2\pi\cos x\sin(2\pi\sin x)$

$g'(x)=0 \Rightarrow \cos x = 0 \ \text{or}\ \sin(2\pi\sin x)=0$

$\cos x = 0 \Rightarrow x = \frac{(2k+1)\pi}{2}$

$\sin(2\pi\sin x)=0 \Rightarrow \sin x = \frac{n}{2} = 0,\ \pm\frac{1}{2},\ \pm1$

$\Rightarrow x = n\pi,\ \frac{\pi}{6}+2k\pi,\ \frac{5\pi}{6}+2k\pi,\ \ldots$

$W = \left\{\frac{m\pi}{2}\right\} \cup \left\{\frac{\pi}{6}+2k\pi,\ \frac{5\pi}{6}+2k\pi,\ \frac{7\pi}{6}+2k\pi,\ \frac{11\pi}{6}+2k\pi\right\}$

$\text{gaps of } W \text{ alternate } \frac{\pi}{3}, \frac{\pi}{6} \Rightarrow \text{NOT an AP} \Rightarrow (r)$

$\frac{\pi}{2},\ \frac{3\pi}{2},\ 4\pi,\ 7\pi \in W \Rightarrow W \supseteq (p)$

$\frac{\pi}{6},\ \frac{7\pi}{6},\ \frac{13\pi}{6} \in W \Rightarrow W \supseteq (s)$

$X \to (p),(q);\quad Y \to (q),(t)$

$Z \to (r);\quad W \to (p),(r),(s)$

$(1)\ (IV)=W:\ (p),(r),(s)\ \checkmark$

$(2),\ (3)\ \text{claim } Z \text{ is AP / contains } (u)\ \times$

$(4)\ \text{claims } W \text{ is AP / contains } (t)\ \times$

$\boxed{X\to(p,q);\ Y\to(q,t);\ Z\to(r);\ W\to(p,r,s)}$

$\boxed{\text{correct option } (1)}$