Question

Consider the following lists:

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

The correct option is:

• A. (I) $\rightarrow$ (P); (II) $\rightarrow$ (S); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (S)
• B. (I) $\rightarrow$ (P); (II) $\rightarrow$ (P); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (R)
• C. (I) $\rightarrow$ (Q); (II) $\rightarrow$ (P); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (S)
• D. (I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (R)

Answer 1

Answer: (B)

(I) $\left\{x \in\left[\frac{-2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$
$\cos x+\sin x=1$
$\Rightarrow \frac{1}{\sqrt{2}} \cos x +\frac{1}{\sqrt{2}} \sin x =\frac{1}{\sqrt{2}} $
$ \Rightarrow \cos \left( x -\frac{\pi}{4}\right)=\cos \frac{\pi}{4} $
$ \Rightarrow x -\frac{\pi}{4}=2 n \pi \pm \frac{\pi}{4} ; n \in Z $
$ \Rightarrow x =2 n \pi ; x =2 n \pi+\frac{\pi}{2} ; n \in Z $
$ \Rightarrow x \in\left\{0, \frac{\pi}{2}\right\} $ in given range has two solutions
(II) $ \left\{ x \in\left[\frac{-5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x =1\right\} $
$ \sqrt{3} \tan 3 x =1 \Rightarrow \tan 3 x =\frac{1}{\sqrt{3}} \Rightarrow 3 x = n \pi+\frac{\pi}{6} $
$ \Rightarrow x =(6 n +1) \frac{\pi}{18} ; n \in Z $
$ \Rightarrow x \in\left\{\frac{\pi}{18}, \frac{-5 \pi}{18}\right\} $ in given range has two solutions
(III) $\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\} $
$ 2 \cos 2 x=\sqrt{3} $
$ \Rightarrow \cos 2 x=\frac{\sqrt{3}}{2}=\cos \frac{\pi}{6}$
$\Rightarrow 2 x =2 n \pi \pm \frac{\pi}{6} ; n \in Z $
$ \Rightarrow x = n \pi \pm \frac{\pi}{12} ; n \in Z $
$ x \in\left\{\pm \frac{\pi}{12}, \pi \pm \frac{\pi}{12},-\pi \pm \frac{\pi}{12}\right\}$
Six solutions in given range
(IV) $ \left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\} $
$ \cos x-\sin x=-1 $
$ \Rightarrow \cos \left(x+\frac{\pi}{4}\right)=\frac{-1}{\sqrt{2}}=\cos \frac{3 \pi}{4} $
$ \Rightarrow x+\frac{\pi}{4}=2 n \pi \pm \frac{3 \pi}{4} ; n \in Z$
$\Rightarrow x=2 n \pi+\frac{\pi}{2} \text { or } x =2 n \pi-\pi ; n \in Z $
$ \Rightarrow x \in\left\{\frac{\pi}{2}, \frac{-3 \pi}{2}, \pi,-\pi\right\} $ four solutions in given range