Question

If $2+3i$ is one of the roots of the equation $2x^3-9x^2+kx-13=0,k\in R$, then the real root of this equation :

• A. exist and is equal to $\frac{1}{2}$
• B. does not exist
• C. exist and is equal to 1
• D. exist and is equal to $-\frac{1}{2}$

Answer 1

The correct option is A exist and is equal to $\frac{1}{2}$

Given equation is
$2x^3-9x^2+kx-13=0$ where $k\in R$
Since, any odd degree equation with real coefficients has at least one real root.
So, real root exist.
Let $\alpha$ be the real root.
Since, $2+3i$ is a root of the equation. So, other root will be 2-3i. (Imaginary roots always occurs in conjugate pairs.)
Now, sum of roots $2+3i+2-3i+\alpha =\frac{9}{2}$
$\Rightarrow \alpha =\frac{1}{2}$