JEE Advanced · Mathematics · 8. Trigonometric Equations

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:
$X = \{x : f (x) = 0\}, Y = \{x : f^{'} (x) = 0\}$
$Z = \{x : g (x) = 0\}, W = \{x : g^{'} (x) = 0\}$
List- $I$ contains the sets $X, Y, Z$ and $W .$ List- $I I$ contains some information regarding these sets.

List- I List- II

$(I)$ $X$ $(P)$ $\supseteq \{\frac{π}{2}, \frac{3 π}{2}, 4 π, 7 π\}$

$(I I)$ $Y$ $(Q)$ an arithmetic progression

$(I I I)$ $Z$ $(R)$ NOT an arithmetic progression

$(I V)$ $W$ $(S)$ $\supseteq \{\frac{π}{6}, \frac{7 π}{6}, \frac{13 π}{6}\}$

$(T)$ $\supseteq \{\frac{π}{3}, \frac{2 π}{3}, π\}$

$(U)$ $\supseteq \{\frac{π}{6}, \frac{3 π}{4}\}$

Which of the following is the only correct combination?

	List- I		List- II
$(I)$	$X$	$(P)$	$\supseteq \{\frac{π}{2}, \frac{3 π}{2}, 4 π, 7 π\}$
$(I I)$	$Y$	$(Q)$	an arithmetic progression
$(I I I)$	$Z$	$(R)$	NOT an arithmetic progression
$(I V)$	$W$	$(S)$	$\supseteq \{\frac{π}{6}, \frac{7 π}{6}, \frac{13 π}{6}\}$
		$(T)$	$\supseteq \{\frac{π}{3}, \frac{2 π}{3}, π\}$
		$(U)$	$\supseteq \{\frac{π}{6}, \frac{3 π}{4}\}$

A $(I I) - (R), (S)$
B $(I) - (P), (R)$
C $(I I) - (Q), (T)$
D $(I) - (Q), (U)$

Verified Solution

Answer & Solution

Correct Answer

(C) $(I I) - (Q), (T)$

Step-by-step Solution

Detailed explanation

(I)

f (x) = \sin (π \cos x)

and

X = \{x : f (x) = 0\}

f (x) = 0

\Rightarrow \sin (π \cos x) = 0

\Rightarrow π \cos x = m π, m \in I

\Rightarrow \cos x = m

\cos x = 0, 1, - 1

\Rightarrow x = \frac{n π}{2}

\Rightarrow X = \{\frac{n π}{2}, n \in I\} \Rightarrow X = \{\pm \frac{π}{2}, \pm π, \pm \frac{3 π}{2}, \pm 2 π \dots \dots\}

(I) - P, Q

(I I) g (x) = \cos (2 π \sin x)

and

Z = \{x : g (x) = 0\}

\cos (2 π \sin x) = 0

\Rightarrow 2 π \sin x = (2 m + 1) \frac{π}{2}, m \in I

\Rightarrow \sin x = \frac{2 n + 1}{4}

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

\Rightarrow Z = \{n π \pm \sin^{- 1} (\frac{1}{4}), n π \pm \sin^{- 1} (\frac{3}{4})\}, n \in I

(I I) - Q, T

(I I I) f (x) = \sin (π \cos x)

and

Y = (x : f^{'} (x) = 0)

f^{'} (x) = \cos (π \cos x) . (- π \sin x) = 0

Now,

\cos (π \cos x) = 0 \Rightarrow π \cos x = (2 m + 1) \frac{π}{2}, m \in I

\Rightarrow \cos x = \frac{2 m + 1}{2}

\Rightarrow \cos x = - \frac{1}{2}, \frac{1}{2} \Rightarrow x = n π \pm \frac{π}{3}, n \in I

\sin x = 0 \Rightarrow x = n π

Hence,

Y = \{n π, n π \pm \frac{π}{3}\}, n \in I

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

(I I I) \to R

(I V)

g (x) = \cos (2 π \sin x)

and

W = \{x : g^{'} (x) = 0\}

g^{'} (x) = - \sin (2 π \sin x) (2 π \cos x) = 0

Now,

\cos x = 0 \Rightarrow x = (2 m + 1) \frac{π}{2}, m \in I

\sin (2 π \sin x) = 0

2 π \sin x = m π, m \in I

\Rightarrow \sin x = \frac{m}{2}

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

\Rightarrow x = \frac{n π}{2}, n π \pm \frac{π}{6}, n \in I

W = \{\frac{n π}{2}, n π \pm \frac{π}{6}\}

W = \{\frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}, π, \frac{7 π}{6}, \frac{3 π}{2} \dots .\}

(I V) - P, R, S

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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