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Question

The equation ex1 + x - 2 = 0 has


A
one real root
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B
two real roots
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C
three real roots
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D
four real roots.
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Solution

The correct option is A one real root
Clearly, x = 1 satisfies the given equation.
Assume that f(x) = ex1 + x  2 = 0 has a real root α other than x = 1. We may suppose
that α > 1 (the case α < 1 is exactly similar).
Applying Rolle's theorem on [1, α] (if α < 1 apply the theorem on[α, 1]), we get β
ϵ (1, α) such that f'(β) = 0.
But f(β) = eβ  1 + 1, so that eβ  1 = 1, which is not possible.
Hence there is no real root other than 1.

Mathematics

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