Question

If the roots of the equation $x^3-3p{x}^2+3qx-r=0$ are in H.P., then the mean root is

• A. $\frac{3r}{q}$
• B. $\frac{2r}{q}$
• C. $\frac{r}{q}$
• D. $-\frac{r}{q}$

Answer 1

Answer: (C)

$\frac{r}{q}$

Let $\alpha ,\beta ,\gamma$ be the roots of $x^3-3p{x}^2+3qx-r=0$
$\alpha +\beta +\gamma =3p\alpha \beta +\beta \gamma +\gamma \alpha =3q\alpha \beta \gamma =r$
As $\alpha ,\beta ,\gamma$ are in H.P
Then
$\beta =\frac{3\alpha \beta \gamma }{\alpha \beta +\beta \gamma +\gamma \alpha }=\frac{3r}{3q}=\frac{r}{q}$