Q65 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics

$\cosh \left(\sinh ^{-1}(\sqrt{8})+\cosh ^{-1} 5\right)=$

  • A
    $\sqrt{6}+4 \sqrt{2}$
  • Correct answer
    B
    $15+8 \sqrt{3}$
  • C
    $6 \sqrt{6}+10 \sqrt{2}$
  • D
    $8-15 \sqrt{3}$

Solution

Since $\cos h\left(\sin h^{-1}(\sqrt{8})+\cos h^{-1}(5)\right)$ $\begin{aligned} & =\cos h[\log (\sqrt{8}+\sqrt{1+8})+\log (5+\sqrt{25-1})] \\ & =\cos h(\log (3+2 \sqrt{2})(5+2 \sqrt{6})) \\ & =\frac{e^{\log (3+2 \sqrt{2})(5+2 \sqrt{6})}+e^{-\log (3+2 \sqrt{2})(5+2 \sqrt{6})}}{2} \\ & =\frac{1}{2}\left\{(3+2 \sqrt{2})(5+2 \sqrt{6})+\frac{1}{(3+2 \sqrt{2})(5+2 \sqrt{6})}\right\} \\ & =\frac{1}{2}\{(3+2 \sqrt{2})(5+2 \sqrt{6})+(3-2 \sqrt{2})(5-2 \sqrt{6})\} \\ & =\frac{1}{2}\{15 \times 2+8 \sqrt{3} \times 2\}=15+8 \sqrt{3}\end{aligned}$

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

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