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Question

Prove that value of λ for which 2x2 - 2(2λ + 1) x + λ (λ + 1) = 0 may have one root less than λ and other root greater than λ are given by λ > 0 or λ < -1.

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Solution

If f (x) = ( - α) (x - β),
then f (λ) = (λ - α) (λ - β) = ive as the two factors are of opposite signs by given conditions.
Also roots must be real
Δ08(λ2+λ+12)
=8[(λ+12)2+14] is +ive, which is true.
Now f (λ) = -ive
2λ22(2λ+1)λ+λ(λ+1)<0
or λ2λ<0orλ(λ+1)>0
or λ<1orλ>0

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