A) \[\frac{10}{17}\]
B) \[-1\]
C) \[-\frac{7}{17}\]
D) \[1\]
Correct Answer: D
Solution :
We have, \[\tan \left( 2{{\tan }^{-1}}\frac{1}{5}-\frac{\pi }{4} \right)\] \[=\tan \left( {{\tan }^{-1}}\frac{5}{12}-{{\tan }^{-1}}1 \right)\] \[\left[ \because 2{{\tan }^{-1}}x={{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}} \right]\] \[=\tan \left\{ {{\tan }^{-1}}\left( \frac{\frac{5}{12}-1}{1+\frac{5}{12}} \right) \right\}=\frac{-7}{17}\] So, the given equation is \[17{{x}^{2}}-7x-10=0\] \[\Rightarrow \] \[(x-1)(17x+10)=0\] \[\Rightarrow \] \[x=1,\frac{-10}{17}\]
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