Q62 / 160 · TS EAMCET 2015 · Mathematics

If $\cosh 2 x=199$, then coth $x$ equals

  • A
    $\frac{5}{3 \sqrt{11}}$
  • B
    $\frac{5}{6 \sqrt{11}}$
  • C
    $\frac{7}{3 \sqrt{11}}$
  • Correct answer
    D
    $\frac{10}{3 \sqrt{11}}$

Solution

We have, $$ \begin{aligned} & \cosh 2 x & =199 \\ \Rightarrow & \frac{1+\tanh ^2 x}{1-\tanh ^2 x} & =199 \\ \Rightarrow & 1+\tanh ^2 x & =199-199 \tanh ^2 x \\ \Rightarrow & 200 \tanh ^2 x & =198 \\ \Rightarrow & \quad \tanh ^2 x & =\frac{198}{200} \Rightarrow \tanh ^2 x=\frac{99}{100} \\ \Rightarrow & & \tanh x=\frac{3 \sqrt{11}}{10} \Rightarrow \operatorname{coth} x=\frac{10}{3 \sqrt{11}} \end{aligned} $$

Hence, the correct answer is option: 'D'.

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