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) | $\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$ | (S) | has five elements | | | | (T) | has six elements | The correct option is:

(B) (I) → (P); (II) → (P); (III) → (T); (IV) → (R)

(A) (I) → (P); (II) → (S); (III) → (P); (IV) → (S)

(C) (I) → (Q); (II) → (P); (III) → (T); (IV) → (S)

(D) (I) → (Q); (II) → (S); (III) → (P); (IV) → (R)

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Trigonometric Equations

Consider the following lists:

The correct option is:

Answer: (B)

(I)

cos x + sin x = 1 \Rightarrow cos (x - π /4) = 1/ 2

, giving

x = 0, π /2

in the range. Two elements → (P)

(II)

3 tan 3 x = 1 \Rightarrow tan 3 x = 1/ 3 \Rightarrow 3 x = nπ + π /6

, so

x = (6 n + 1) π /18

. In

[- 5 π /18, 5 π /18]

x = - 5 π /18, π /18

. Two elements → (P)

(III)

2 cos 2 x = 3 \Rightarrow cos 2 x = 3 /2 \Rightarrow 2 x = 2 nπ \pm π /6

, so

x = nπ \pm π /12

. In

[- 6 π /5, 6 π /5]

x = \pm π /12, π \pm π /12, - π \pm π /12

. Six elements → (T)

(IV)

sin x - cos x = 1 \Rightarrow cos (x + π /4) = - 1/ 2

, giving

x = π /2, - 3 π /2, π, - π

in the range. Four elements → (R)

Correct matching: (I)→(P), (II)→(P), (III)→(T), (IV)→(R), which is option (B).

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Single correct option