JEE Main · 2010 · 11 Apr 2010 · Offline · Trigonometric Ratios and Identities

For a regular polygon the inradius and circumradius satisfy $\frac{r}{R}=\cos \frac{\pi }{n}$, so a ratio is achievable only if it equals $\cos \frac{\pi }{n}$ for some integer $n\ge 3$. The value $\frac{1}{\sqrt{2}}$ gives $n=4$, $\frac{\sqrt{3}}{2}$ gives $n=6$ and $\frac{1}{2}$ gives $n=3$, all valid. But $\frac{r}{R}=\frac{2}{3}$ would need $\cos \frac{\pi }{n}=\frac{2}{3}\approx 0.667$, which lies between the values for $n=3$ and $n=4$ and matches no integer $n$. So the false claim is the one asserting a regular polygon with $\frac{r}{R}=\frac{2}{3}$.