Silvia made a $$15,000$ interest$-$free loan to her son, Alex, who used the money to buy a used car. Alex's only sources of income was wages from his job, and he had no investment income. The relevant Federal interest rate $($AFR$)$ was $3\%.$ Based on the above information:

a$)$ Alex would have deemed interest expense of $$450(\mathrm{.}03$$$15,000).$

b$)$ Silvia must recognize $$450(\mathrm{.}03$$$15,000)$ of imputed interest income on the below$-$market loan.

c$)$ Alex's gross income must be increased by the $$450(\mathrm{.}03$$$15,000)$ imputed interest income on the below market loan.

$$ d$)$ Silvia does not recognize any imputed interest income and Alex does not recognize any imputed interest expense.