MCQ
If $A, B, C$ are acute positive angles such that $A + B + C = \pi $ and $\cot A\,\cot \,B\,\cot \,C = K,$ then
- ✓$K \le \frac{1}{{3\sqrt 3 }}$
- B$K \ge \frac{1}{{3\sqrt 3 }}$
- C$K < \frac{1}{9}$
- D$K > \frac{1}{3}$
$ \Rightarrow \tan A + \tan B + \tan C = \tan A\tan B\tan C$
Now $A.M.$ $\geq$ $G.M. $
$ \Rightarrow \frac{{\tan A + \tan B + \tan C}}{3} \ge {(\tan A\tan B\tan C)^{1/3}}$
$ \Rightarrow \left( {\frac{{\tan A\tan B\tan C}}{3}} \right) \ge {(\tan A\tan B\tan C)^{1/3}}$
$ \Rightarrow {(\tan A\tan B\tan C)^{2/3}} \ge 3$
$ \Rightarrow {\left( {\frac{1}{K}} \right)^{2/3}} \ge 3$
$\Rightarrow \frac{1}{K} \ge {3^{3/2}} $
$\Rightarrow K \le \frac{1}{{3\sqrt 3 }}$.
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