JEE Advanced 2022 — Paper 2 — MATH-II solution

JEE Advanced2022Paper 2MATH-II

Let $P QR S$ be a quadrilateral in a plane, where $QR = 1$ , $∠ P QR = ∠ QR S = 7 0^{\circ}$ , $∠ P QS = 1 5^{\circ}$ and $∠ P R S = 4 0^{\circ}$ . If $∠ R P S = θ^{\circ}$ , $P Q = α$ and $P S = β$ , then the interval(s) that contain(s) the value of $4 α β sin θ^{\circ}$ is/are

A
$(0, 2)$
B
$(1, 2)$
C
$(2, 3)$
D
$(22, 32)$

Solution

Applying sine rule in $△ P QR$ ,

$\dfrac{\alpha}{\sin 30^\circ} = \dfrac{1}{\sin 80^\circ}$

$\Rightarrow \alpha = \dfrac{1}{2\sin 80^\circ} \qquad \cdots (i)$

Applying sine rule in $△ P R S$ ,

$\dfrac{\beta}{\sin 40^\circ} = \dfrac{1}{\sin \theta}$

$\Rightarrow \beta\sin\theta = \sin 40^\circ \qquad \cdots (ii)$

From (i) and (ii),

$4\alpha\beta\sin\theta = 4\cdot \dfrac{1}{2\sin 80^\circ} \cdot \sin 40^\circ = \dfrac{1}{\cos 40^\circ}$

Using $cos 0^{\circ} = 1$ , $cos 3 0^{\circ} = \frac{3}{2}$ , $cos 4 5^{\circ} = \frac{1}{2}$ and $cos 6 0^{\circ} = \frac{1}{2}$ .

Only Options (A) and (B) are correct.

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