IIT-JEE 2011 · Paper 2, Mathematics question 42

Common root α. Subtracting the equations: $(b-1)\alpha-(1+b)=0$, so $\alpha=\dfrac{1+b}{b-1}$.

Substitute into $\alpha^2+\alpha+b=0$ and multiply by $(b-1)^2$:
$$(1+b)^2+(1+b)(b-1)+b(b-1)^2=b(b^2+3)=0.$$
So $b=0$ or $b=\pm i\sqrt3$. Among the options, $-i\sqrt3$.

(B)