Question

If the $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+b{x}^2+3x-1=0(\alpha \le \beta \le \gamma ,\alpha ,\beta ,\gamma$ are in $H.P$.) then

• A. One of the roots must be $1$.
• B. One root is smaller than $1$, other is greater then $1$.
• C. $b\in [-3,\infty )$.
• D. All the roots must be equal.

Answer 1

Answer: (A), (D)

One of the roots must be $1$.
All the roots must be equal.

$\alpha ,\beta ,\gamma$ are in H.P.
$\Rightarrow \beta =\frac{2\alpha \gamma }{\alpha +\gamma }$
$\beta \alpha +\beta \gamma -2\alpha \gamma =0.....\left(A\right)$
$x^3+b{x}^2+3x-1=0$
$\alpha +\beta +\gamma =-b$ .....(1)
$\alpha \beta +\beta \gamma +\gamma \alpha =3$ ....(2)
$\alpha \beta \gamma =1$ .....(3)
From (A) and (2), $\Rightarrow \gamma \alpha =1$
From (3), $\beta =1$ and $\alpha =\gamma =\beta =1$.