Answer
(a) $b\gt 10$;
(b) $b$ does not exist;
(c) $b\leq10$.
Work Step by Step
Given $a=10, B=150^\circ$, use the Law of Sines, we have $\frac{sinB}{b}=\frac{sinA}{a}$ or $b=\frac{a\ sinB}{sinA}=\frac{10sin150^\circ}{sinA}=\frac{5}{sinA}$ or $sinA=\frac{5}{b}$, based on the figure, we have:
(a) For $A$ has exactly one value, we need $b\gt 10$;
(b) For $A$ has two possible values, value $b$ does not exist;
(c) For $A$ has no value, that is no triangle can be formed, we have $b\leq10$.
