Question

If $2$ and $3$ are the roots of cubic equation $2x^3+m{x}^2-13x+n=0$, then the values of $m$ and $n$ are:

• A. $-5,-30$
• B. $-5,30$
• C. $5,30$
• D. $Noneofthese$

Answer 1

The correct option is B $-5,30$

$if\alpha ,\beta ,\gamma$are roots of a cubic equation then,
cubic equation is written as;
$x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\alpha \gamma )x-\alpha \beta \gamma =0$
Given, $2$ and $3$ are roots of the equation $2x^3+m{x}^2-13x+n=0$
Let $\alpha$ be the third root .
$5+\alpha =-\frac{m}{2}$ ....(1)
Also, $6+2\alpha +3\alpha =-\frac{13}{2}$
$\Rightarrow \alpha =-\frac{5}{2}$
Also, $6\alpha =-\frac{n}{2}$
$\Rightarrow n=30$
Now, by (1), $m=-5$