Christine purchased a $$6,000$ bond that was paying a $5\mathrm{.}50\%$ compounded semi$-$annually coupon rate and had $7$ more years to maturity. The yield rate at the time of purchase was $4\mathrm{.}75\%$ compounded semi$-$annually.

a$.$ How much did Christine pay for the bond?

b$.$ What was the amount of premium or discount on the bond?