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4-2.Quadratic Equations and Inequations
hard
The number of real roots of the equation ${e^{\sin x}} - {e^{ - \sin x}} - 4$ $ = 0$ are
A
$1$
B
$2$
C
Infinite
D
None
(IIT-1982)
Solution
(d) Given equation ${e^{\sin x}} – {e^{ – \sin x}} – 4 = 0$
Let ${e^{\sin x}} = y$, then given equation can be written as
${y^2} – 4y – 1 = 0$==> $y = 2 \pm \sqrt 5 $
But the value of $y = {e^{\sin x}}$ is always positive, so
$y = 2 + \sqrt 5 \,\,\,(2 < \sqrt 5 )$
==> ${\log _e}y = {\log _e}(2 + \sqrt 5 )$
==>$\sin x = {\log _e}(2 + \sqrt 5 ) > 1$
which is impossible, since $\sin x$ cannot be greater than $1$. Hence we cannot find any real value of $x $ which satisfies the given equation.
Standard 11
Mathematics
