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

Answer 1

The correct option is C exists and is equal to $\frac{1}{2}$

Whenever the coefficients of a polynomial are real, complex roots exist in conjugate pair.
Here, the coefficients are real and one of the root is $2+3i$, So the other root must be $2-3i$. Let the other root be $\alpha .$

Sum of the roots $=2+3i+2-3i+\alpha =\frac{9}{2}\Rightarrow \alpha =\frac{1}{2}$