John borrows $1,00$o from Jane at an annual effective rate of interest i$.$ He agrees to pay back $1,00$o after six years and $1,366\mathrm{.}87$ after another six years.

Three years after his first payment, John repays the outstanding balance.

What is the amount of John's second payment?

$($A$)1,020$

$($B$)1,027$

$($C$)1,048$

$($D$)1,073$

$($E$)1,094$

Fund A is invested at an effective annual interest rate of $3\%.$

Fund B is invested at an effective annual interest rate of $2\mathrm{.}5\%.$

At the end of $20$ years, the total in the two funds is $10,000.$ At the end of $31$ years, the amount in Fund A is twice the amount in Fund B$.$

Calculate the total in the two funds at the end of $10$ years.

$($A$)5,732$

$($B$)6,602$

$($C$)7,472$

$($D$)7,569$

$($E$)8,123$

You are given:

$($i$)$ The sum of the present values of a payment of X at the end of $10$ years and a payment of $Y$ at the end of $20$ years is equal to the present value of a payment of $x+Y$ at the end of $15$ years.

$($ii$)x+Y=100$

$($iii$)i=5\%$

Calculate X$.$

$($A$)44$

$($B$)48$

$($C$)50$

$($D$)52$

$($E$)57$

$7.$ John is $30$ years old. He will receive $2$ payments of $$2,500$ each. The first payment will be an unknown number of years in the future. The second payment will be five years after the first payment. At an annual effective interest rate of $i=5\%,$ the present value of the two payments is $$2,607.$

Determine at what age John will receive the second payment.

$($A$)$ Less than $40$

$($B$)$ At least $40$ but less than $45$

$($C$)$ At least $45$ but less than $50$

$($D$)$ At least $50$ but less than $55$

$($E$)$ At least $55$

$8.$ Carl puts $10,000$ into a bank account that pays an annual effective interest rate of $4\%$ for ten years. If a withdrawal is made during the first five and one$-$half years, a penalty of $5\%$ of the withdrawal amount is made. Carl withdraws K at the end of each of years $4,5,6$ and $7.$ The balance in the account at the end of year $10$ is $10,000.$

Calculate K $.$

$($A$)929$

$($B$)958$

$($C$)980$

$($D$)1,005$

$($E$)1\mathrm{.}031$