2022 YIJC P1 Q10

Alan and Betty bought an apartment at \(\$450 \; 000\). They are eligible to take a housing loan, up to \(85\%\) of the cost of the apartment, for a maximum of \(30\) years.

After careful consideration, the couple decides to borrow \(85\%\) of the cost of the apartment. They will make a cash repayment of \(\$x\) at the beginning of each month, starting \(1\)st July \(2022\). Interest will be charged with effect from \(31\)st July \(2022\) at a monthly interest rate of \(0.2\%\) for the remaining amount owed at the end of each month.

1. Find the amount of money owed on \(31\)st of July \(2022\) after the interest for the month has been added. Express your answer in terms of \(x\).

2. Show that the total amount of money owed after the \(n\) th repayment at the beginning of the month is

   [1]
   \({{1.002}^{n-1}}\left( \mathrm{382500} \right)-500x\left( {{1.002}^{n}}-1 \right)\)[4]
3. Find the earliest date on which the couple will be able to pay off the loan completely if \(x=2000\), and state the amount of repayment on this date.[4]

4. If the couple wishes to pay off the loan completely on \(1\)st Jan \(2050\) (after the repayment on this day), what should the monthly repayment be?

   [3]