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4-2.Quadratic Equations and Inequations
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The number of real roots of the equation $\mathrm{e}^{4 \mathrm{x}}-\mathrm{e}^{3 \mathrm{x}}-4 \mathrm{e}^{2 \mathrm{x}}-\mathrm{e}^{\mathrm{x}}+1=0$ is equal to $.....$
A
$7$
B
$2$
C
$3$
D
$4$
(JEE MAIN-2021)
Solution
$t^{4}-t^{3}-4 t^{2}-t+1=0, e^{x}=t>0$
$\Rightarrow t^{2}-t-4-\frac{1}{t}+\frac{1}{t^{2}}=0$
$\Rightarrow \alpha^{2}-\alpha-6=0, \alpha=t+\frac{1}{t} \geq 2$
$\Rightarrow \alpha=3,-2(\text { reject })$
$\Rightarrow t+\frac{1}{t}=3$
$\Rightarrow$ The number of real roots $=2$
Standard 11
Mathematics
