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(A) $(I) \{x \in[-\frac{2 \pi}{3}, \frac{2 \pi}{3}]: \cos x+\sin x=1\}$$\cos x+\sin x=1$ $\Rightarrow \frac{1}{\sqrt{2}} \cos x+\frac{1}{\sqrt{2}} \si

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$(I)$ $\rightarrow (P); (II)$ $\rightarrow (P); (III)$ $\rightarrow (T); (IV)$ $\rightarrow (R)$

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(A) $(I) \{x \in[-\frac{2 \pi}{3}, \frac{2 \pi}{3}]: \cos x+\sin x=1\}$$\cos x+\sin x=1$ $\Rightarrow \frac{1}{\sqrt{2}} \cos x+\frac{1}{\sqrt{2}} \sin x=\frac{1}{\sqrt{2}}$ $\Rightarrow \cos(x-\frac{\pi}{4})=\cos \frac{\pi}{4}$$x-\frac{\pi}{4}=2n\pi \pm \frac{\pi}{4} \Rightarrow x=2n\pi, 2n\pi+\frac{\pi}{2}$. In range $[-\frac{2\pi}{3}, \frac{2\pi}{3}]$,$x \in \{0, \frac{\pi}{2}\}$. Two elements $ightarrow (P)$.$(II) \{x \in[-\frac{5 \pi}{18}, \frac{5 \pi}{18}]: \sqrt{3} 	an 3 x=1\}$$	an 3x = \frac{1}{\sqrt{3}}$ $\Rightarrow 3x = n\pi + \frac{\pi}{6}$ $\Rightarrow x = \frac{n\pi}{3} + \frac{\pi}{18}$.For $n=0, x=\frac{\pi}{18}$. For $n=-1, x=-\frac{5\pi}{18}$. For $n=1, x=\frac{7\pi}{18} > \frac{5\pi}{18}$. Two elements $ightarrow (P)$.$(III) \{x \in[-\frac{6 \pi}{5}, \frac{6 \pi}{5}]: 2 \cos 2x = \sqrt{3}\}$$\cos 2x = \frac{\sqrt{3}}{2}$ $\Rightarrow 2x = 2n\pi \pm \frac{\pi}{6}$ $\Rightarrow x = n\pi \pm \frac{\pi}{12}$.Solutions: $\pm \frac{\pi}{12}, \pi \pm \frac{\pi}{12}, -\pi \pm \frac{\pi}{12}$. Total six elements $ightarrow (T)$.$(IV) \{x \in[-\frac{7 \pi}{4}, \frac{7 \pi}{4}]: \sin x-\cos x=1\}$$\cos x - \sin x = -1 \Rightarrow \cos(x+\frac{\pi}{4}) = -\frac{1}{\sqrt{2}} = \cos \frac{3\pi}{4}$.$x+\frac{\pi}{4} = 2n\pi \pm \frac{3\pi}{4} \Rightarrow x = 2n\pi + \frac{\pi}{2}$ or $x = 2n\pi - \pi$.For $n=0, x=\frac{\pi}{2}, -\pi$. For $n=1, x=\frac{5\pi}{2} (	ext{out}), \pi$. For $n=-1, x=-\frac{3\pi}{2}, -3\pi (	ext{out})$.Solutions: $\{\frac{\pi}{2}, -\pi, \pi, -\frac{3\pi}{2}\}$,four elements $ightarrow (R)$.

$List-I$	$List-II$
$(I)$ $\{x \in[-\frac{2 \pi}{3}, \frac{2 \pi}{3}]: \cos x+\sin x=1\}$	$(P)$ has two elements
$(II)$ $\{x \in[-\frac{5 \pi}{18}, \frac{5 \pi}{18}]: \sqrt{3} \tan 3 x=1\}$	$(Q)$ has three elements
$(III)$ $\{x \in[-\frac{6 \pi}{5}, \frac{6 \pi}{5}]: 2 \cos (2 x)=\sqrt{3}\}$	$(R)$ has four elements
$(IV)$ $\{x \in[-\frac{7 \pi}{4}, \frac{7 \pi}{4}]: \sin x-\cos x=1\}$	$(S)$ has five elements
	$(T)$ has six elements

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