$[9$ marks$]$ A $$1,150$ callable bond pays interest at j$\_(2)=10\%$ matures on November $26,2032.$ It can be called annually, from $2025$ to $2032($inclusive$),$ on November $26$ at par. a$.$ Find the purchase price on November $26,2021,$ to guarantee an investor a yield of $8\%$ compounded semi$-$annually. b$.$ Suppose the bond was purchased on November $26,2021$ for the price determined in part $($a$).$ If the bond is not called $($i$.$e$.$ held to maturity$),$ determine the purchaser's annual yield compounded semi$-$annually $($i$.$e$.$ determine j$\_(2))$ using: i$)$ the method of averages. ii$)$ the method of interpolation.