A) 0
B) 1
C) 2
D) infinitely many
Correct Answer: A
Solution :
We have, the given equation as \[\sin \,({{e}^{x}})={{5}^{x}}+{{5}^{-x}}\] ?.(i) Let \[{{5}^{x}}=t,\]then Eq. (i), reduces to \[\sin \,({{e}^{x}})=t+\frac{1}{t}\] \[\Rightarrow \] \[\sin \,({{e}^{x}})=t+\frac{1}{t}-2+2\] \[\Rightarrow \] \[\sin \,({{e}^{x}})=\left( \sqrt{t}-\frac{1}{\sqrt{t}} \right)+2\] \[(\because \,{{5}^{x}}>0,\therefore \sqrt{{{5}^{x}}}=\sqrt{t}\,\text{exists})\] \[\Rightarrow \] \[\sin ({{e}^{x}})\ge 2\] which is not possible as \[\sin \theta \le 1\] Thus, given equation has no solution.
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