Question

lf ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are the roots of the equation $3x^4-(l+m)x^3+2x+5l=0$ and sum of all the roots is equal to $3$ and ${\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4=10$, then $(l,m)$ is equal to

• A. $(6,3)$
• B. $(-6,-3)$
• C. $(-6,15)$
• D. $(6,15)$

Answer 1

The correct option is A $(6,3)$

If ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are the roots of the equation $3x^4-(l+m)x^3+2x+5l=0$
then, ${\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4=(l+m)/3=3$ (given) .....(1)
And ${\alpha }_1\cdot {\alpha }_2\cdot {\alpha }_3\cdot {\alpha }_4=5l/3=10$ (given)
$\Rightarrow l=6$ and using (1), $m=9-l=3$