Let f ( x ) = sin ( cos x ) and g ( x ) = cos ( 2 sin x ) be two functions defined for x 0 . Define the following sets whose elements are written in the increasing order. Column - I contains the sets X, Y, Z and W. Column - II contains some informati

Question

Let $f (x) = \sin (π \cos x)$ and $g (x) = \cos (2 π \sin x)$ 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. [2019]

	Column I		Column II
(I)	X	(p)	$\supseteq {\frac{π}{2}, \frac{3 π}{2}, 4 π, 7 π}$
(II)	Y	(q)	an arithmetic progression
(III)	Z	(r)	NOT an arithmetic progression
(IV)	W	(s)	$\supseteq {\frac{π}{6}, \frac{7 π}{6}, \frac{13 π}{6}}$
		(t)	$\supseteq {\frac{π}{3}, \frac{2 π}{3}, π}$
		(u)	$\supseteq {\frac{π}{6}, \frac{3 π}{4}}$

Which of the following is the only CORRECT combination?

Smart Achievers · Accepted Answer

(4)

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

$x > 0$

$\Rightarrow x = \frac{π}{2}, π, \frac{3 π}{2}, 2 π, \frac{5 π}{2}, 3 π, \frac{7 π}{2}, 4 π, \dots$

$∴$ $X = {\frac{π}{2}, π, \frac{3 π}{2}, 2 π, \frac{5 π}{2}, 3 π, \frac{7 π}{2}, 4 π, \dots}$

$∴$ $(I) \to P, Q$

$f' (x) = 0 \Rightarrow \cos (π \cos x) (- π \sin x) = 0$

$\Rightarrow \cos (π \cos x) = 0, \sin x = 0$

$\Rightarrow π \cos x = (2 n - 1) \frac{π}{2}, x = n π$

$\Rightarrow \cos x = \frac{2 n - 1}{2}, x = π, 2 π, 3 π, \dots$

$\Rightarrow \cos x = \frac{- 1}{2}, \frac{1}{2}$

$\Rightarrow x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}, \frac{7 π}{3}, \frac{8 π}{3}, \frac{10 π}{3}, \frac{11 π}{3}, \frac{13 π}{3}, \dots$

$∴$ $Y = {\frac{π}{3}, \frac{2 π}{3}, π, \frac{4 π}{3}, \frac{5 π}{3}, 2 π, \dots}$

$∴$ $(I I) \to Q, T$

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

$\Rightarrow 2 π \sin x = (2 n - 1) \frac{π}{2}$ $\Rightarrow \sin x = \frac{2 n - 1}{4}$

$\Rightarrow \sin x = \frac{1}{4}, - \frac{1}{4}, \frac{3}{4}, - \frac{3}{4}$

$∴$ $Z = {- \sin^{- 1} \frac{1}{4}, - \sin^{- 1} \frac{3}{4}, \sin^{- 1} \frac{1}{4}, \sin^{- 1} \frac{3}{4}}$

$∴$ $(I I I) \to R$

$g' (x) = 0 \Rightarrow - \sin (2 π \sin x) \cdot 2 π \cos x = 0$

$\Rightarrow \sin (2 π \sin x) = 0, \cos x = 0$

$\Rightarrow 2 π \sin x = n π, x = (2 n - 1) \frac{π}{2}$

$\Rightarrow \sin x = \frac{n}{2}, x = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2}, \dots$

$\Rightarrow \sin x = - 1, - \frac{1}{2}, 0, \frac{1}{2}, 1$

$\Rightarrow x = \frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}, π, \frac{3 π}{2}, \frac{11 π}{6}, 2 π, \frac{13 π}{6}, \dots$

$∴$ $W = {\frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}, π, \frac{3 π}{2}, \frac{11 π}{6}, 2 π, \frac{13 π}{6}, \dots}$

$∴$ $(I V) \to P, R, S$

Solution

1 (I), (q), (u)

2 (I), (p), (r)

3 (II), (r), (s)

4 (II), (q), (t)

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