A $1-$year T$-$note, face value $$100,$ paying a coupon rate of $8\%$ p$.$a$.,$ trading at a yield to maturity of $6\%$ p$.$a$.($semi annual compounding$).$

$($a$)$ Explain briefly why relying on the above information alone, there is no way one can figure out the price of a one$-$year zero coupon bond with a face value of $$100.$

$($b$)$ In addition to the above information, it is given that a $0\mathrm{.}5$ year $T-$note, paying a coupon rate of $4\%$ p$.$a$.,$ is trading at par. Compute the price of a one$-$year zero coupon bond with a face value of $$100.$

$($c$)$ Based on the given information and your work in part b$),$ what is the price of a one$-$year T$-$note, with a face value of $$100,$ paying a coupon rate of $4\%$ p$.$a$.?$

Show the necessary computation.