JEE MAIN - Mathematics (2021 - 27th July Evening Shift - No. 18)
The number of real roots of the equation e4x $$-$$ e3x $$-$$ 4e2x $$-$$ ex + 1 = 0 is equal to ______________.
Answer
2
Explanation
t4 $$-$$ t3 $$-$$ 4t2 $$-$$ t + 1 = 0, ex = t > 0
$$ \Rightarrow {t^2} - t - 4 - {1 \over t} + {1 \over {{t^2}}} = 0$$
$$ \Rightarrow {\alpha ^2} - \alpha - 6 = 0,\alpha = t + {1 \over t} \ge 2$$
$$ \Rightarrow \alpha = 3, - 2$$ (reject)
$$ \Rightarrow t + {1 \over t} = 3$$
$$\Rightarrow$$ The number of real roots = 2
$$ \Rightarrow {t^2} - t - 4 - {1 \over t} + {1 \over {{t^2}}} = 0$$
$$ \Rightarrow {\alpha ^2} - \alpha - 6 = 0,\alpha = t + {1 \over t} \ge 2$$
$$ \Rightarrow \alpha = 3, - 2$$ (reject)
$$ \Rightarrow t + {1 \over t} = 3$$
$$\Rightarrow$$ The number of real roots = 2
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