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Question

# For the equation x2 - (a - 3) x + a = 0 (a ∈ R), find the values of 'a' such that exactly one root lies in between 1 and 2.

A

a (10, )

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B

a (-, -10)

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C

a (-, -1)

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D

None of these

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Solution

## The correct option is A a (10, ) Consider the graph of f(x) = x2 - (a - 3) x + a For roots to be real D ≥ 0 b2 - 4ac ≥ 0 ⇒ a less than or equal to 1 (or) a ≥ 9 ----------- (1) For exactly 1 root to lie in between 1 and 2, f(1) and f(2) should be of opposite signs. f(1) f(2) < 0 f(x) = x2 - (a - 3) x + a (given) f(1) = 1 - (a - 3) + a = 4 f(2) =4 - 2(a - 3) + a = 4 - 2a + 6 + a = 10 - a f(1) (2) < 0 4 (10 - a) < 0 a ϵ (10, ∞) ----------------- (2) From (1) & (2), a ϵ (10, ∞)

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