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1991

JEE Main 1991 Mathematics question on Trigonometry

The number of solutions of the equation sin⁡(ex)=5x+5−x\sin(e^x) = 5^x + 5^{-x} is
  1. A.
    0
  2. B.
    1
  3. C.
    2
  4. D.
    infinitely many
Show answer and solution

Correct option: A

Solution

The left side sin(eˣ) is bounded between -1 and 1. The right side 5ˣ + 5⁻ˣ ≥ 2√(5ˣ · 5⁻ˣ) = 2 by AM-GM inequality, with equality never achieved for real x (it equals 2 only in the limit, never attained). Since the right side is always ≥ 2 > 1 ≥ left side, there are no real solutions.

[1991]

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