Q105 / 160 · TS EAMCET 2017 · Mathematics

If $\cosh ^{-1} x=2 \log _e(\sqrt{2}+1)$, then $x=$

  • A
    1
  • B
    2
  • C
    4
  • Correct answer
    D
    3

Solution

We have, $\cosh ^{-1} x=2 \log _e(\sqrt{2}+1)$ $$ \begin{aligned} & \Rightarrow \log _e\left(x+\sqrt{x^2-1}\right)=\log _e(\sqrt{2}+1)^2 \\ & \Rightarrow \quad x+\sqrt{x^2-1}=(\sqrt{2}+1)^2 \\ & \Rightarrow \quad x+\sqrt{x^2-1}=3+2 \sqrt{2}=3+\sqrt{8} \end{aligned} $$ On comparing rational and irrational part, we get $$ x=3 $$

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

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