$,$ if you have purchased a $10$ year bond issued by MTR company to finance the future railway development on $1$ April $2024.$ The bond has a face value of $$1,000,$ carries $8\%$ coupon and it paid semi$-$annually. The yield of the bond is $10\%($APR$)$ compounded semi$-$annually.

$($a$)$if you pay to buy the bond on $1$ April $2024,$ how much is it$?$

$($b$)$ Is the bond a par bond, discount bond or premium bond? Explain. $($c$)$ On $31$ March $2025,$ the bond becomes a par bond as its yield has decreased from $($APR$)10\%$

to $($APR$)$ X$\%,$ compounded semi$-$annually.

$($i$)$ What is the value of X$?$ Explain.

$($ii$)$ If you sell the bond immediately after receiving the $($second$)$ coupon, calculate the capital gain yield.