If the roots of the equation x^3 - ax^2 + bx | vedclass

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Question

(C) Let the roots of the cubic equation be $\frac{p}{r}, p, pr$. From the relation between roots and coefficients: $1) \frac{p}{r} + p + pr = a \Right

Vedclass · Accepted Answer

$c$

Vedclass · Answer

(C) Let the roots of the cubic equation be $\frac{p}{r}, p, pr$. From the relation between roots and coefficients: $1) \frac{p}{r} + p + pr = a \Rightarrow p(\frac{1}{r} + 1 + r) = a$ $2) \frac{p}{r} \cdot p + p \cdot pr + pr \cdot \frac{p}{r} = b \Rightarrow p^2(\frac{1}{r} + r + 1) = b$ $3) \frac{p}{r} \cdot p \cdot pr = p^3 = c$ Dividing equation $(2)$ by equation $(1)$: $\frac{p^2(\frac{1}{r} + r + 1)}{p(\frac{1}{r} + r + 1)} = \frac{b}{a} \Rightarrow p = \frac{b}{a}$ Substituting $p = \frac{b}{a}$ into equation $(3)$: $(\frac{b}{a})^3 = c \Rightarrow \frac{b^3}{a^3} = c$ Thus,the correct option is $C$.

If the roots of the equation $x^3 - ax^2 + bx - c = 0$ are in $GP$,then $\frac{b^3}{a^3}$ is equal to:

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