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 | Column II | | --- | --- | | (I) X | (p) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ | | (II) Y | (q) an arithmetic progression | | (III) Z | (r) NOT an arithmetic progression | | (IV) W | (s) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ | | | (t) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ | | | (u) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$ | Which of the following is the only CORRECT combination?

$f(x) = \sin(\pi \cos x), \qquad g(x) = \cos(2\pi \sin x), \quad x > 0$

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

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

$\text{Match } X, Y, Z, W \text{ with the correct statement } (\text{p})\text{--}(\text{u}).$

$\text{Choose the only CORRECT combination.}$

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

$\Rightarrow \cos x = n, \quad n \in \mathbb{Z}$

$\cos x \in \{-1, 0, 1\} \Rightarrow x = \frac{n\pi}{2}, \quad n = 1, 2, 3, \dots$

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

$\text{common difference} = \frac{\pi}{2} \ \Rightarrow \ X \text{ is an A.P. } (q)$

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

$X \longrightarrow (p), \ (q)$

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

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

$-\pi\sin x\,\cos(\pi\cos x) = 0$

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

$x = n\pi \ \text{ or } \ \cos x = \frac{2k+1}{2} = \pm\frac{1}{2}$

$Y = \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}, 2\pi, \dots\right\}$

$= \left\{\frac{n\pi}{3}\right\}, \ n = 1, 2, 3, \dots \ \Rightarrow \ \text{A.P. } (q)$

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

$Y \longrightarrow (q), \ (t)$

$g(x) = 0 \Rightarrow \cos(2\pi\sin x) = 0$

$2\pi\sin x = \frac{(2k+1)\pi}{2} \Rightarrow \sin x = \pm\frac{1}{4}, \ \pm\frac{3}{4}$

$\text{gaps are not constant} \Rightarrow Z \text{ is NOT an A.P. } (r)$

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

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

$-2\pi\cos x\,\sin(2\pi\sin x) = 0$

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

$\sin x = \frac{n}{2} \Rightarrow \sin x \in \left\{0, \pm\frac{1}{2}, \pm 1\right\}$

$W = \left\{x : \sin x \in \left\{0, \pm\tfrac{1}{2}, \pm 1\right\}\right\}$

$\text{residues (units of } \tfrac{\pi}{6}\text{)} = \{0,1,3,5\} \ \Rightarrow \text{NOT an A.P. } (r)$

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

$\begin{array}{c|c} \text{Set} & \text{Column II} \\ X & (p),\ (q) \\ Y & (q),\ (t) \\ Z & (r) \\ W & (r),\ (s) \end{array}$

$\text{The only correct combination is option } (2)$

boxed(I) X → (p), (III) Z → (r)