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VITEEE 2010 Paper

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Question : 84 of 120

Marks: +1, -0

The number of solutions of the equation sin

(e^{x})=5^{x}+5^{-x},

is

0
1
2
infinitely many

Solution:

We have, sin

(e^{x})=5^{x}+5^{-x},

...(i)

Let

5^{x}=t,

then eq. (i), reduces to sin

(e^{x})=t+\frac{1}{t}

⇒ sin

(e^{x})=t+\frac{1}{t} -2+2

⇒ sin

(e^{x})=\left(\sqrt{t}-\frac{1}{\sqrt{t}}\right)^{2} + 2

(\because\;\;5^{x}>0,\therefore\sqrt{5^{x}}=\sqrt{t}\;\;\text{exists})

⇒ sin

(e^{x})\geq 2

which is not possible as also sinθ ≤ 1.

Thus, given equation has no solution.

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