Question

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

• A. 0
• B. 1
• C. 2
• D. infinitely many

Answer 1

The Correct Option is A

We have, the given equation as
$\sin \left(e^{x}\right)=5^{x}+5^{-x}$ ...(i)
Let $5^{x}=t$, then E (i), reduces to
$\sin \left(e^{x}\right)=t+\frac{1}{t}$
$\Rightarrow \sin \left(e^{x}\right)=t+\frac{1}{t}-2+2$
$\Rightarrow \sin \left(e^{x}\right)=\left(\sqrt{t}-\frac{1}{\sqrt{t}}\right)^{2}+2$
$\left\{\because 5^{x}>0, \therefore \sqrt{5^{x}}=\sqrt{t} \text { exists }\right\}$
$\Rightarrow \sin \left(e^{x}\right) \geq 2$
which is not possible as $\sin \theta \leq 1$.
Thus, given equation has no solution.