The smallest positive integral value of a, fo | vedclass

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(D) $x^4 - ax^2 + 9 = 0$ . . . . $(1)$ Let $x^2 = t$. Then $t^2 - at + 9 = 0$ . . . . $(2)$ For the roots of equation $(1)$ to be real and distinct, t

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(D) $x^4 - ax^2 + 9 = 0$ . . . . $(1)$ Let $x^2 = t$. Then $t^2 - at + 9 = 0$ . . . . $(2)$ For the roots of equation $(1)$ to be real and distinct, the roots of equation $(2)$ must be positive and distinct. $(i)$ Discriminant $D > 0$ $\Rightarrow a^2 - 36 > 0$ $\Rightarrow a \in (-\infty, -6) \cup (6, \infty)$. $(ii)$ Sum of roots $\frac{-b}{a} > 0 \Rightarrow a > 0$. $(iii)$ Product of roots $\frac{c}{a} > 0 \Rightarrow 9 > 0$, which is true for all $a \in \mathbb{R}$. By taking the intersection of $(i), (ii),$ and $(iii)$, we get $a > 6$. Therefore, the smallest positive integral value of $a$ is $7$.

The smallest positive integral value of $a$, for which all the roots of $x^4 - ax^2 + 9 = 0$ are real and distinct, is equal to

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