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Question

The number of real solutions of the equation $$\sin(\mathrm{e}^{\mathrm{x}})=5^{\mathrm{x}}+5^{-x}$$ is (are) 


A
0
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B
1
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C
2
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D
Infinitely many
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Solution

The correct option is A $$0$$
$$\sin (e^{x})$$ has a max value of 1.

$$5^{x}+5^{-x}$$ has a max value of 2.

$$A.M\geq G.M$$

$$\Rightarrow \dfrac{5^{x}+5^{-x}}{2}\geq \sqrt{5^{x}.5^{-x}}$$

$$5^{x}+5^{-x}\geq 2.$$

$$\therefore $$ There is no point of intersection of both.

$$\therefore $$ No solution.

Mathematics

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