If \tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}, then x is

NIMCET 2009 Question Paper with solutions

Question : 7 of 120

Marks: +1, -0

\tan ^{-1} 2 x+\tan ^{-1} 3 x=\;{π}/{4}

, then

x

Solution:

\tan ^{-1} 2 x+\tan ^{-1} 3 x=\;{π}/{4}

⇒ \;\; \tan ^{-1}(\;{2 x+3 x}/{1-2 x ⋅ 3 x})=\;{π}/{4}

⇒ \;\; \tan ^{-1} \;{5 x}/{1-6 x^{2}}=\;{π}/{4}

⇒ \;\; \;{5 x}/{1-6 x^{2}}=\tan \;{π}/{4}=1

\Rightarrow

1-6 x^{2}=5 x

\Rightarrow

6 x^{2}+5 x-1=0

(6 x-1)(x+1)=0

⇒ \;\; x=\;{1}/{6}

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