Q68 / 160 · TS EAMCET 2023 (12 May Shift 2) · Mathematics

$\sinh (\log (3+\sqrt{8}))=$

  • A
    $3^{\frac{3}{2}}$
  • Correct answer
    B
    $2^{\frac{3}{2}}$
  • C
    $8^{\frac{2}{3}}$
  • D
    $3^{\frac{1}{2}}$

Solution

$\begin{aligned} & \quad \sin h(\log (3+\sqrt{8})) \\ & =\sin h(\log (3+2 \sqrt{2}))=\sin h\left\{\log (\sqrt{2}+1)^2\right\} \\ & \because \sin h x=\frac{e^x-e^{-x}}{2} \\ & \sin h\left(\log (\sqrt{2}+1)^2\right)=\frac{e^{\log (\sqrt{2}+1)^2}-e^{-\log (\sqrt{2}+1)^2}}{2} \\ & =\frac{(\sqrt{2}+1)^2-\frac{1}{(\sqrt{2}+1)^2}}{2} \\ & =\frac{(\sqrt{2}+1)^2-(\sqrt{2}-1)^2}{2}=2 \sqrt{2}=2^{3 / 2} .\end{aligned}$

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

Relevant Questions