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Question

The values of a for which 2x2 - 2 (2a + 1)x + a (a + 1) = 0 may have one root less than a and other root greater than a are given by


A

1 > a > 0

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B

-1 < a < 0

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C

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D

a > 0 or a < -1

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Solution

The correct option is D

a > 0 or a < -1


The given condition suggest that a lies between the roots.

Let f(x) =2x22(2a+1)x+a(a+1)

For ‘a’ to lie between the roots we must have Discriminant 0 and f(a) < 0

Now, Discriminant 0

4(2a+1)2 – 8a(a + 1) 0

8(a2+a+12)0 which is always true

Also f(a) < 0 2a2 – 2a(2a + 1) + a(a + 1) < 0

a2 – a < 0 a2 + a > 0 a(1 + a) > 0

a > 0 or a < –1


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