A) \[\lambda >27\]
B) \[\lambda <27\]
C) \[\lambda >3\]
D) \[\lambda <3\]
Correct Answer: A
Solution :
[a]\[{{\left( \tan A-\tan B \right)}^{2}}+{{\left( \tan B-\tan C \right)}^{2}}+{{(\tan C+\tan A)}^{2}}>0\]\[\Rightarrow \,\,2\left( {{\tan }^{2}}A+{{\tan }^{2}}B+{{\tan }^{2}}C-\tan A\tan B-\tan B\tan C-\tan C\tan A \right)>0\]\[\Rightarrow \,\,3\left( {{\tan }^{2}}A+{{\tan }^{2}}B+{{\tan }^{2}}C \right)-{{\left( \tan A+\tan B+\tan C \right)}^{2}}>0\]\[\Rightarrow \,\,\,3\lambda -{{\left( \tan A+\tan B+\tan C \right)}^{2}}>0\] \[\Rightarrow \,\,\,3\lambda -81>0\] \[\Rightarrow \,\,\,\lambda >27\]
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