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(D) Given that $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in a geometric progression $(GP)$. For three terms $x, y, z$ to be i

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$c\Delta = 0$

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(D) Given that $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in a geometric progression $(GP)$. For three terms $x, y, z$ to be in $GP$,$y^2 = xz$. Therefore,$(\alpha^2 + \beta^2)^2 = (\alpha + \beta)(\alpha^3 + \beta^3)$. Expanding both sides: $\alpha^4 + \beta^4 + 2\alpha^2\beta^2 = \alpha^4 + \alpha\beta^3 + \beta\alpha^3 + \beta^4$. Simplifying: $2\alpha^2\beta^2 = \alpha\beta(\alpha^2 + \beta^2)$. Rearranging: $\alpha\beta(\alpha^2 + \beta^2 - 2\alpha\beta) = 0$. This gives $\alpha\beta(\alpha - \beta)^2 = 0$. Since $\alpha\beta = \frac{c}{a}$ and $(\alpha - \beta)^2 = \frac{(\alpha + \beta)^2 - 4\alpha\beta}{1} = \frac{b^2 - 4ac}{a^2} = \frac{\Delta}{a^2}$. Substituting these values: $\frac{c}{a} 	imes \frac{\Delta}{a^2} = 0$. Thus,$c\Delta = 0$.

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in a geometric progression,and $\Delta = b^2 - 4ac$,then which of the following is true?

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