The value of a for which the quadratic equati | vedclass

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(C) For a quadratic equation $Ax^2 + Bx + C = 0$ to have roots with opposite signs,the product of the roots must be negative.The product of the roots

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$(1, 2)$

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(C) For a quadratic equation $Ax^2 + Bx + C = 0$ to have roots with opposite signs,the product of the roots must be negative.The product of the roots is given by $\frac{C}{A}$. Here,$A = 3$ and $C = a^2 - 3a + 2$.Thus,we require $\frac{a^2 - 3a + 2}{3} Factoring the quadratic expression,we get $(a - 1)(a - 2) This inequality holds when $1 Additionally,for the roots to be real,the discriminant $D = B^2 - 4AC$ must be greater than or equal to $0$. Since the product of the roots is negative,the roots are guaranteed to be real and distinct,so the discriminant condition is automatically satisfied for all $a$ in the interval $(1, 2)$.

The value of $a$ for which the quadratic equation $3x^2 + 2(a^2 + 1)x + (a^2 - 3a + 2) = 0$ possesses roots with opposite signs,lies in

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