Question

Let $\alpha +i\beta ,\alpha ,\beta ϵR$ be a root of the equation $x^3+qx+r=0,q,rϵR$. The cubic equation is independent of $\alpha and\beta$ whose one root is $2\alpha$, is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$\alpha \pm i\beta$ be complex roots of equation $x^3+qx+r=0$ and γ be its real root.
$\Rightarrow (x-\gamma )(x^2-2\alpha x+{\alpha }^2+{\beta }^2)=x^3+qx+r$
On comparing coefficients, we get
$-\gamma -2\alpha =0\Rightarrow \gamma =-2\alpha$
Since, ${\gamma }^3+q\gamma +r=0$
$\Rightarrow (-2\alpha {)}^3+q(-2\alpha )+r=0$
i.e., $2\alpha$ is root of cubic equation $x^3+qx-r=0$