If the roots of the cubic equation ax^3 + bx^ | vedclass

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(A) Let the roots of the equation $ax^3 + bx^2 + cx + d = 0$ be $\frac{A}{R}, A, AR$ where $A$ and $R$ are constants.Since the roots are in $G.P.$,the

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$c^3a = b^3d$

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(A) Let the roots of the equation $ax^3 + bx^2 + cx + d = 0$ be $\frac{A}{R}, A, AR$ where $A$ and $R$ are constants.Since the roots are in $G.P.$,their product is $\frac{A}{R} \cdot A \cdot AR = A^3 = -\frac{d}{a}$.Thus,$A = -\left(\frac{d}{a}ight)^{1/3}$.Since $A$ is a root of the equation,it must satisfy $aA^3 + bA^2 + cA + d = 0$.Substituting $A^3 = -\frac{d}{a}$ and $A = -\left(\frac{d}{a}ight)^{1/3}$:$a\left(-\frac{d}{a}ight) + b\left(-\frac{d}{a}ight)^{2/3} + c\left(-\frac{d}{a}ight)^{1/3} + d = 0$.$-d + b\left(\frac{d}{a}ight)^{2/3} - c\left(\frac{d}{a}ight)^{1/3} + d = 0$.$b\left(\frac{d}{a}ight)^{2/3} = c\left(\frac{d}{a}ight)^{1/3}$.Cubing both sides:$b^3 \cdot \frac{d^2}{a^2} = c^3 \cdot \frac{d}{a}$.Multiplying both sides by $a^2$:$b^3d^2 = c^3ad$.Dividing by $d$ (assuming $d 
eq 0$):$b^3d = c^3a$.

If the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$ are in $G.P.$,then

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