Question

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

• A. $\frac{3n}{m}$
• B. $\frac{2n}{m}$
• C. $\frac{n}{m}$
• D. $-\frac{n}{m}$

Answer 1

Answer: (A)

$\frac{3n}{m}$

As roots are in H.P the let $\alpha ,\beta ,\gamma$ roots of $x^3-l{x}^2+mx-n=0$
$S_1=\alpha +\beta +\gamma =l{S}_2=\alpha \beta +\beta \gamma +\gamma \alpha =m{S}_3=\alpha \beta \gamma =n$
Now as $\alpha ,\beta ,\gamma$ are in H.P
Then $\frac{1}{\alpha },\frac{1}{\beta },\frac{1}{\gamma }$ are in A.P
Hence,
Mean of A.P is $\frac{\alpha \beta +\beta \gamma +\gamma \alpha }{3\alpha \beta \gamma }$
So mean of H.P is $\frac{3\alpha \beta \gamma }{\alpha \beta +\beta \gamma +\gamma \alpha }=\frac{3n}{m}$