QJEEAdv22Ma389a: JEE Advanced 2022 Mathematics - Class 11 JEEAdv22Ma389a Solution: Let be a quadrilateral in a plane, where and . If and , then…

Question

Verified Solution • 2025–26 Session moderate

Let

P Q R S

be a quadrilateral in a plane, where

Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}

and

\angle P R S=40^{\circ}

.

If

\angle R P S=\theta^{\circ}, P Q=\alpha

and

P S=\beta

, then the interval(s) that contain(s) the value of

4 \alpha \beta \sin \theta^{\circ}

is/are

Accepted Answer

Applying sine rule in

\triangle P Q R

,

\begin{aligned} \frac{α}{\sin 30^{\circ}} = \frac{1}{\sin 80^{\circ}} \\ \Rightarrow & α = \frac{1}{2 \sin 80^{\circ}} \end{aligned}

Applying sine rule in

\triangle P R S

,

\begin{aligned} \frac{β}{\sin 40^{\circ}} = \frac{1}{\sin θ} \\ \Rightarrow β \sin θ = & \sin 40^{\circ} \end{aligned}

From (i) and (ii),

4 α β \sin θ = 4 \cdot \frac{1}{2 \sin 80^{\circ}} \cdot \sin 40^{\circ} = \frac{1}{\cos 40^{\circ}}

Using

\cos 0^{\circ}=1, \cos 30^{\circ}=\frac{\sqrt{3}}{2}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}

and

\cos 60^{\circ}=\frac{1}{2}

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

Step-by-Step Verified Solution