Which of the following is true for any value of $a$ in the expressions, for at least one negative $x$ .

(a^{2} + 3) x^{2} + (a + 2) x - 6 > 0

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(a^{2} + 3) x^{2} + (a + 2) x - 6 < 0

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(a^{2} + 3) x^{2} + (a + 2) x - 6 = 0

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None of these.

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Solution

The correct option is A $(a^{2} + 3) x^{2} + (a + 2) x - 6 > 0$
Let $y = (a^{2} + 3) x^{2} + (a + 2) x - 6$
Discriminant is, $Δ = (a + 2)^{2} + 24 (a^{2} + 3) > 0 \forall a \in R$
Also coefficient of $x^{2}$ is $a^{2} + 3 > 0 \forall a \in R$
Hence $y > 0 \forall x \in R & \forall a \in R$

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