IB DP Maths- MAI HL Prediction Paper 1 – 2026 Edition
IB DP Math AI HL Prediction Paper 1 – April/May 2026 Exam
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Question 1: Financial Applications of Geometric Sequences and Series [15 marks]
Imani invests $3000 in a bank that pays a nominal annual interest rate of 1.25% compounded monthly.
a Question 1a [4 marks] – Future Value of Investment
Calculate the amount of money Imani will have in the bank at the end of 6 years:
Show Solution
Using the compound interest formula
\( FV = PV \left(1 + \frac{r}{n} \right)^{nt} \)
Substituting \( PV = 3000 \), \( r = 0.0125 \), \( n = 12 \), \( t = 6 \)
\( FV = 3233.53 \)
Detailed Solution:
- Formula: \( FV = PV \left(1 + \frac{r}{n} \right)^{nt} \).
- Given:
- \( PV = 3000 \) (initial investment).
- \( r = 1.25\% = 0.0125 \) (annual interest rate).
- \( n = 12 \) (compounded monthly).
- \( t = 6 \) years.
- Calculate:
- \( FV = 3000 \left(1 + \frac{0.0125}{12} \right)^{12 \times 6} \).
- \( FV = 3000 \left(1.00104167 \right)^{72} \).
- \( FV = 3233.53 \).
- Result: The amount is $3233.53.
b Question 1b [4 marks] – Time to Reach $3550
Calculate the number of months it takes until Imani has at least $3550 in the bank:
Show Solution
Using the compound interest formula
\( 3550 = 3000 \left(1 + \frac{0.0125}{12} \right)^{12t} \)
Solving for \( t \)
\( t \approx 162 \) months
Detailed Solution:
- Formula: \( FV = PV \left(1 + \frac{r}{n} \right)^{nt} \).
- Given: \( FV = 3550 \), \( PV = 3000 \), \( r = 0.0125 \), \( n = 12 \).
- Setup: \( 3550 = 3000 \left(1 + \frac{0.0125}{12} \right)^{12t} \).
- Simplify: \( \frac{3550}{3000} = \left(1.00104167 \right)^{12t} \).
- Logarithms:
- \( \log \left( \frac{3550}{3000} \right) = 12t \log \left(1.00104167 \right) \).
- \( t = \frac{\log \left( \frac{3550}{3000} \right)}{12 \log \left(1.00104167 \right)} \).
- \( t \approx 13.4738 \) years.
- Convert to months: \( 13.4738 \times 12 \approx 161.686 \).
- Result: Since it’s “at least” $3550, round up to 162 months.
c Question 1c [2 marks] – Loan Amount
Imani uses the \$3550 as a partial payment for a used car costing \$22000. Write down the amount of money that Imani takes out as a loan:
Show Solution
Subtracting partial payment from car cost
\( \text{Loan} = 22000 – 3550 \)
\( \text{Loan} = 18450 \)
Detailed Solution:
- Car cost: $22000.
- Partial payment: $3550.
- Calculate: \( 22000 – 3550 = 18450 \).
- Result: The loan amount is $18450.
d Question 1d [5 marks] – Monthly Loan Payment
The loan is for 8 years and the nominal annual interest rate is 12.6% compounded monthly. Imani will pay the loan in fixed monthly installments at the end of each month. Calculate the amount, correct to the nearest dollar, that Imani will have to pay the bank each month:
Show Solution
Using the loan payment formula
\( PMT = \frac{PV \times \frac{r}{n}}{1 – (1 + \frac{r}{n})^{-nt}} \)
Substituting \( PV = 18450 \), \( r = 0.126 \), \( n = 12 \), \( t = 8 \)
\( PMT = 306 \)
Detailed Solution:
- Formula: \( PMT = \frac{PV \times \frac{r}{n}}{1 – (1 + \frac{r}{n})^{-nt}} \).
- Given:
- \( PV = 18450 \) (loan amount).
- \( r = 12.6\% = 0.126 \) (annual interest rate).
- \( n = 12 \) (monthly compounding).
- \( t = 8 \) years.
- Calculate:
- Monthly rate: \( \frac{0.126}{12} = 0.0105 \).
- Number of payments: \( 12 \times 8 = 96 \).
- Numerator: \( 18450 \times 0.0105 \).
- Denominator: \( 1 – (1 + 0.0105)^{-96} \).
- \( PMT = \frac{18450 \times 0.0105}{1 – (1.0105)^{-96}} \).
- \( PMT \approx 306 \) (rounded to the nearest dollar).
- Result: The monthly payment is $306.