NIMCET 2016 — Mathematics PYQ

Concept:
The internal angle of the polygon of side n,  
$\theta =\frac{n-2}{n}\pi$

Calculation:  
Internal angle of 12 sided polygon ($\theta$) = $\left(\frac{12-2}{12}\right)\pi$  

$\Rightarrow \theta =\frac{5}{6}\pi =150^{\circ }$

Given OC (radius) = 3 cm  
$\angle OCB=\theta /2=75^{\circ }$

In right angle triangle, OCB  
$BC=OC\cos (\angle OCB)$  

$\Rightarrow BC=3\cos 75^{\circ }$  

$\Rightarrow BC=3\cos (45^{\circ }+30^{\circ })$  

$\Rightarrow BC=3(\cos 45^{\circ }\cos 30^{\circ }-\sin 45^{\circ }\sin 30^{\circ })$  

$\Rightarrow BC=3\left(\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\cdot \frac{1}{2}\right)$  

Now the side of the polygon $AB=2BC$

$\Rightarrow AB=2\times \frac{3}{2\sqrt{2}}(\sqrt{3}-1)$  

$\Rightarrow AB=\frac{3}{\sqrt{2}}(\sqrt{3}-1)$  

\Rightarrow AB \approx 1.553 < 3

∴ AB < 3 and only one option $18-9\sqrt{3}$ is positive and < 3, so the option is selected.