e^{{sech}^{-1} 1/2 + ^{-1} 1/2 + ^{-1} 1/2} =

Correct Answer is : 3+23+35+2152{3+2 {3}+3 {5}+2 {15}}{2}23+23+35+215

Correct Answer is : 3+23+35+2152{3+2 {3}+3 {5}+2 {15}}{2}23+23+35+215

Test Index

AP EAMCET Engineering 21 Apr 2019 Shift 2 Paper

Section: Mathematics

Question : 25 of 160

Marks: +1, -0

e^{\operatorname{sech}^{-1} \frac{1}{2} + \tanh^{-1} \frac{1}{2} + \sinh^{-1} \frac{1}{2}}

Solution:

We know that,

\operatorname{sech}^{-1} x = \log \left( \frac{1+\sqrt{1-x^{2}}}{x} \right)

\tanh^{-1} x = \frac{1}{2} \log \left( \frac{1+x}{1-x} \right)

\sinh^{-1} x = \log \left( x + \sqrt{x^{2}+1} \right)

Substitute the values in the given equation and simplify.

e^{ \log \left( \frac{1+\sqrt{1-\frac{1}{4}}}{\frac{1}{2}} \right) + \frac{1}{2} \log \left( \frac{1+\frac{1}{2}}{1-\frac{1}{2}} \right) + \log \left( \frac{1}{2} + \sqrt{\frac{1}{4}} + 1 \right) } = (2+\sqrt{3})(\sqrt{3})\left( \frac{\sqrt{5}+1}{2} \right)

= \frac{(2 \sqrt{3}+3)(\sqrt{5}+1)}{2}

= \frac{2 \sqrt{15}+2 \sqrt{3}+3 \sqrt{5}+3}{2}

= \frac{3+3 \sqrt{5}+2 \sqrt{3}+2 \sqrt{15}}{2}

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