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Q. $ f ( x )=0$ is a cubic equation with positive and distinct roots $\alpha, \beta, \gamma$ such that $\beta$ is harmonic mean between the roots of $f ^{\prime}( x )=0$. If $r =\left[\frac{2 \beta}{\alpha+\gamma}\right]+\left[\frac{2 \alpha \gamma}{\alpha \beta+\beta \gamma}\right]$, then the value of $\displaystyle\sum_{ i =1}^3( i )^{ r }$ is
$[$ Note $:[k]$ denotes greatest integer function less than or equal to $k$. $]$

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Solution:

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$\beta=\frac{2 x_1 x_2}{x_1+x_2}=\frac{2 b}{-2 a}=\frac{b}{-a}=\frac{-(\alpha \beta+\beta \gamma+\gamma \alpha)}{-(\alpha+\beta+\gamma)} $
$\alpha \beta+\beta^2+\beta \gamma=\alpha \beta+\beta \gamma+\gamma \alpha$
$\beta^2=\alpha \gamma \Rightarrow \alpha, \beta, \gamma \text { are in G.P. }$
$r=\left[\frac{G \cdot M}{\text { A.M. }}\right]+\left[\frac{\text { H.M. }}{\text { G.M. }}\right]=0+0=0$
$\displaystyle\sum_{i=I}^3 1=1+1+1=3$