The values of a for which 2x^2 - 2(2a + 1)x + | vedclass

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(D) Let $f(x) = 2x^2 - 2(2a + 1)x + a(a + 1)$.For one root to be less than $a$ and the other root to be greater than $a$,the value of the quadratic fu

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$a > 0 	ext{ or } a < -1$

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(D) Let $f(x) = 2x^2 - 2(2a + 1)x + a(a + 1)$.For one root to be less than $a$ and the other root to be greater than $a$,the value of the quadratic function at $x = a$ must be negative,i.e.,$f(a) Substituting $x = a$ into the function:$f(a) = 2(a)^2 - 2(2a + 1)(a) + a(a + 1)$$f(a) = 2a^2 - 4a^2 - 2a + a^2 + a$$f(a) = -a^2 - a$Setting $f(a) $-a^2 - a $a^2 + a > 0$$a(a + 1) > 0$This inequality holds when $a > 0$ or $a Since the discriminant $D = 4(2a + 1)^2 - 8a(a + 1) = 8(a^2 + a + 1/2) = 8((a + 1/2)^2 + 1/4) > 0$ for all real $a$,the roots are always real. Thus,the condition $f(a) < 0$ is sufficient.

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

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