$A$ is reducible to $B$ implies $B$ is as tough as $A$. ($A$ cannot be harder than $B$)
Option $A$ - False. If $A$ is polynomial then $B$ must be Polynomial ($A$ polynomial algorithm can be easily converted into exponential. Converse is not true).
Option $B$- True. As per first line above, if $A$ is expositional then $B$ cannot be a polynomial time.
Option $C$ - False. If we have polynomial time algorithm for $A$ then we can have polynomial, expositional, sub expositional algorithm for $B$.
Option $D$- False. $A$ can be polynomial. $B$ can be harder than polynomial ( as per first line, $B$ is as hard as $A$ can be termed as $B=\Omega(A)$).