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4-2.Quadratic Equations and Inequations
hard
The number of solutions, of the equation $\mathrm{e}^{\sin x}-2 e^{-\sin x}=2$ is
A
$2$
B
more than $2$
C
$1$
D
$0$
(JEE MAIN-2024)
Solution
Take $e^{\sin x}=t(t>0)$
$\Rightarrow \mathrm{t}-\frac{2}{\mathrm{t}}=2$
$\Rightarrow \frac{\mathrm{t}^2-2}{\mathrm{t}}=2$
$\Rightarrow \mathrm{t}^2-2 \mathrm{t}-2=0$
$\Rightarrow \mathrm{t}^2-2 \mathrm{t}+1=3$
$\Rightarrow(\mathrm{t}-1)^2=3$
$\Rightarrow \mathrm{t}=1 \pm \sqrt{3}$
$\Rightarrow \mathrm{t}=1 \pm 1.73$
$\Rightarrow \mathrm{t}=2.73 \text { or }-0.73(\text { rejected as } \mathrm{t}>0)$
$\Rightarrow \mathrm{e}^{\sin \mathrm{x}}=2.73$
$\Rightarrow \log _{\mathrm{e}} \mathrm{e}^{\sin \mathrm{x}}=\log _{\mathrm{e}} 2.73$
$\Rightarrow \sin \mathrm{x}=\log _{\mathrm{e}} 2.73>1$
So no solution.
Standard 11
Mathematics
