Hwa Chong Institution H2 Math papers, with video solutions

Jane deposited \( \$x\) on 1 January \(2018\) and then a further \( \$x\) on 1 February \(2018\) and at the start of every subsequent month in a bank account. The interest rate was \(0.3\%\) per month, so that on the last day of each month the amount in the account on that day was increased by \(0.3\%\).

1. Find an expression in terms of \(n\) and \(x\) for the amount of money in the bank account on the last day of the \(n\) th month.[3]
2. What is the minimum amount (to the nearest dollar) Jane should deposit in the bank every month if she were to have at least \( \$5000\) in the bank at the end of \(15\) months?[2]

After \(2\) years, Jane stopped depositing \( \$x\) per month, and instead decided to withdraw \( \$200\) from the account on the \({{15}^{\mathrm{th}}}\) of every subsequent month, starting on \({{15}^{\mathrm{th}}}\) January 2020. The interest rate remained as \(0.3\%\) per month, so that on the last day of each month the amount in the account on that day was increased by \(0.3\%\).

3. Using the value of \(x\) found in part (ii), find the date in which Jane would do her last withdrawal till she has no more money left in her account, and the amount of her last withdrawal to the nearest cent.[5]

Having no money left in her bank account, Jane decided to take an interest free loan of \( \$10\,000\) from a friend. She would repay her friend \( \$100\) on the last day of the first month she took the loan. On the last day of each subsequent month, she would repay her friend \( \$150\) more than the amount she repaid in the previous month.

4. Find the number of months Jane will take to repay her loan.[2]

A couple takes up housing loan of $\$L$ and the interest is charged before each monthly repayments at a fixed rate of $p\%$ per annum. Their monthly repayment commences on 1 September 2021. Monthly repayments of $\$x$ are due and payable on the first day of subsequent months until their housing loan is fully repaid.

1. State an expression in terms of $L$ and $p$ for the interest charged before their first repayment on 1 September 2021.

   [1]

2. Show that the outstanding loan at the start of the ${{n}^{\mathrm{th}}}$ month after their monthly repayment is given by
   ${{\left( 1+\frac{p}{1200} \right)}^{n}}L-\frac{1200x}{p}\left[ {{\left( 1+\frac{p}{1200} \right)}^{n}}-1 \right]$.

   [3]

The couple is taking a housing loan of $\$504,000$ at a fixed interest rate of $2.6\%$ per annum.

3. Calculate the monthly repayment if the couple plans to repay the loan in $30$ years.

   [2]

Expand $\frac{{{x}^{2}}+2x}{2{{x}^{2}}+1}$ in ascending powers of $x$ up to and including the term in ${{x}^{5}}$. State the range of values of $x$ for which this expansion is valid.

[3]

Find, in the simplest form, the coefficient of ${{x}^{2017}}$ in this expansion.

[2]