Question

Let $PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS={70}^{\circ },\angle PQS={15}^{\circ }$ and $\angle PRS={40}^{\circ }$.
If $\angle RPS={\theta }^{\circ },PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of $4\alpha \beta \sin {\theta }^{\circ }$ is/are

• A. $\left(C\right)(\sqrt{2},3)$
• B. $\left(B\right)(1,2)$
• C. $\left(D\right)(2\sqrt{2},3\sqrt{2})$
• D. $\left(A\right)(0,\sqrt{2})$

Answer 1

Answer: (B), (D)

$\left(A\right)(0,\sqrt{2})$

Figure as shown, can be drawn on the basis of given data.

Applying sine rule in $△PQR,$
$\frac{\alpha }{\sin {30}^{\circ }}=\frac{1}{\sin {80}^{\circ }}$
$\Rightarrow \alpha =\frac{1}{2\sin {80}^{\circ }}$
Applying sine rule in $△PRS$
$\frac{\beta }{\sin {40}^{\circ }}=\frac{1}{\sin \theta }$
$\Rightarrow \beta \sin \theta =\sin {40}^{\circ }$
From (i) & (ii),
$4\alpha \beta \sin \theta =4\cdot \frac{1}{2\sin {80}^{\circ }}\cdot \sin {40}^{\circ }=\frac{1}{\cos {40}^{\circ }}$

${30}^{\circ }<{40}^{\circ }<{45}^{\circ }\Rightarrow \cos {45}^{\circ }<\cos {40}^{\circ }<\cos {30}^{\circ }\Rightarrow \frac{1}{\sqrt{2}}<\cos {40}^{\circ }<\frac{\sqrt{3}}{2}\Rightarrow \frac{2}{\sqrt{3}}<\frac{1}{\cos {40}^{\circ }}<\sqrt{2}\Rightarrow \frac{2}{\sqrt{3}}<4\alpha \beta \sin {\theta }^{\circ }<\sqrt{2}$