The equation e^{ x} - e^{- x} - 4 = 0 has: [AIEEE 2012]

Correct Answer is : no real roots

Correct Answer is : no real roots

Test Index

Quadratic Equation and Inequalities Part 1

Section: Mathematics

Question : 75 of 100

Marks: +1, -0

The equation

e^{\sin x} - e^{-\sin x} - 4 = 0

Solution:

Given equation is

e^{\sin x} - e^{-\sin x} - 4 = 0

Put

e^{\sin x} = t

in the given equation,

we get

t^2 - 4t - 1 = 0

\Rightarrow t = \frac{4 \pm \sqrt{16+4}}{2}

= \frac{4 \pm \sqrt{20}}{2}

= \frac{4 \pm 2\sqrt{5}}{2}

= 2 \pm \sqrt{5}

\Rightarrow e^{\sin x} = 2 \pm \sqrt{5}

t = e^{\sin x})

\Rightarrow e^{\sin x} = 2 - \sqrt{5}

and

e^{\sin x} = 2 + \sqrt{5}

\Rightarrow e^{\sin x} = 2 - \sqrt{5} < 0

\text{ and } \sin x = \ln(2+\sqrt{5}) > 1 \text{ so, rejected }

Hence given equation has no solution.

\therefore

The equation has no real roots.

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