The values of a for which the equation 2x^2 - | vedclass

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(D) Let $f(x) = 2x^2 - 2(2a + 1)x + a(a + 1)$.For $a$ to lie between the roots of the quadratic equation $f(x) = 0$,the condition is $f(a) < 0$.Substi

Vedclass · Accepted Answer

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

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(D) Let $f(x) = 2x^2 - 2(2a + 1)x + a(a + 1)$.For $a$ to lie between the roots of the quadratic equation $f(x) = 0$,the condition is $f(a) Substituting $x = a$ into $f(x)$:$f(a) = 2(a)^2 - 2(2a + 1)(a) + a(a + 1) $f(a) = 2a^2 - 4a^2 - 2a + a^2 + a $-a^2 - a Multiplying by $-1$ (reversing the inequality sign):$a^2 + a > 0$$a(a + 1) > 0$This inequality holds when $a > 0$ or $a Since the discriminant $D = [2(2a+1)]^2 - 4(2)(a(a+1)) = 4(4a^2 + 4a + 1) - 8(a^2 + a) = 16a^2 + 16a + 4 - 8a^2 - 8a = 8a^2 + 8a + 4 = 8(a^2 + a + 1/2) = 8((a + 1/2)^2 + 1/4) > 0$ for all real $a$,the roots are always real and distinct. Thus,the condition $f(a) < 0$ is sufficient.

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

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