Question
The number of solutions of the equation $$\sin {\left( e \right)^x} = {5^x} + {5^{ - x}}\,\,{\text{is}}$$
A.
0
B.
1
C.
2
D.
Infinitely many
Answer :
0
Solution :
The given eq. is $$\sin {\left( e \right)^x} = {5^x} + {5^{ - x}}$$
We know $${5^x}$$ and $${5^{ - x}}$$ both are $$+ve$$ real numbers using
$$\eqalign{
& {\text{AM}} \geqslant {\text{GM}} \cr
& \therefore \,\,{{\text{5}}^x} + \frac{1}{{{5^x}}} \geqslant 2 \cr
& \Rightarrow \,\,{5^x} + {5^{ - x}} \geqslant 2 \cr
& \therefore \,\,{\text{R}}{\text{.H}}{\text{.S of given eq}}{\text{. }} \geqslant {\text{2}} \cr
& {\text{While }}\sin {e^x} \in \left[ { - 1,1} \right] \cr
& {\text{i}}{\text{.e}}{\text{. L}}{\text{.H}}{\text{.S }} \in \left[ { - 1,1} \right] \cr
& \therefore \,\,{\text{The equation is not possible for any real value of }}\,x. \cr
& {\text{Hence}}\,\,{\text{(A)}}\,\,{\text{is the correct answer}}{\text{.}} \cr} $$