$\backslash (3\mathrm{.}1\mathrm{.}14$: $\backslash )$ A lender wishes to obtain a minimum average annual interest rate of $\backslash (10\backslash \%\backslash )$ on a loan to a particular risk class of borrowers. Determine the interest rate that must be charged on loans to members of that risk class in the following cases. $($a$)$ One year loan and on average $\backslash (2\backslash \%\backslash )$ of borrowers default at the end of the year. $($b$)$ One year loan and on average $\backslash (1\backslash \%\backslash )$ of borrowers default in full and $\backslash (1\backslash \%\backslash )$ of borrowers have a $\backslash (50\backslash \%\backslash )$ default $($repay half of the full amount due$).($c$)$ Two year amortized loan with level payments and on average $\backslash (1\backslash \%\backslash )$ of borrowers default at the end of the first year and $\backslash (1\backslash \%\backslash )$ default at the end of the second year.