Question

$\mathrm{If}\mathrm{f}\left(\mathrm{x}\right)=\cosh \mathrm{x}+\sinh \mathrm{x}\mathrm{and}\mathrm{f}\left(\mathrm{p}\right)=\mathrm{f}\left(\mathrm{x}\right).\mathrm{f}\left(\mathrm{y}\right)$ then the value of $p$is

• A. $\mathrm{xy}$
• B. $\mathrm{x}-\mathrm{y}$
• C. $\mathrm{x}+\mathrm{y}$
• D. None of these

Answer 1

The correct option is C

$\mathrm{x}+\mathrm{y}$

Explanation for the correct options:

Determination of given expression:

$$\begin{gathered} \mathrm{f}\left(\mathrm{y}\right)=\cosh \mathrm{y}+\sinh \mathrm{y} \\ \mathrm{now}, \\ \mathrm{f}\left(\mathrm{x}\right).\mathrm{f}\left(\mathrm{y}\right)=(\cosh \mathrm{x}+\mathrm{sinhx}).(\cosh \mathrm{y}+\sinh \mathrm{y}) \\ =\cosh \mathrm{x}\cosh \mathrm{y}+\cosh \mathrm{x}\sinh \mathrm{y}+\sinh \mathrm{x}\cosh \mathrm{y}+\sinh \mathrm{x}\sinh \mathrm{y} \\ =(\cosh \mathrm{x}\cosh \mathrm{y}+\sinh \mathrm{x}\sinh \mathrm{y})+(\sinh \mathrm{x}\cosh \mathrm{y}+\cosh \mathrm{x}\sinh \mathrm{y}) \\ =\cosh (\mathrm{x}+\mathrm{y})+\sinh (\mathrm{x}+\mathrm{y}) \\ =\mathrm{f}(\mathrm{x}+\mathrm{y}) \\ \mathrm{from}\mathrm{given}, \\ \mathrm{f}\left(\mathrm{p}\right)=\mathrm{f}\left(\mathrm{x}\right).\mathrm{f}\left(\mathrm{y}\right) \\ \mathrm{p}=\mathrm{x}+\mathrm{y} \end{gathered}$$

Hence, the correct option is option (C).