Question

The roots of the equation $x^3-14x^2+56x-64=0$ are in

• A. $AGP$
• B. $HP$
• C. $AP$
• D. $GP$

Answer 1

The correct option is D $GP$

Let $\alpha ,\beta ,\gamma$ the roots of $x^3-14x^2+56x-64=0$
$\alpha +\beta +\gamma =14\alpha \beta +\beta \gamma +\gamma \alpha =56\alpha \beta \gamma =64$
Now let roots are in G.P, such that $\alpha \gamma ={\beta }^2$
$\alpha \beta \gamma =64\Rightarrow {\beta }^3=64\Rightarrow \beta =4\Rightarrow \alpha \gamma =16$
And $\alpha \beta +\beta \gamma +\gamma \alpha =\beta (\alpha +\gamma )+16=\beta (14-\beta )+16=4\left(10\right)+6=56$
Therefore, roots are in $GP$
Hence, option 'D' is correct.