IB Maths AI HL · Unit 2: Functions
IB Maths AI HL Advanced Modelling Questions
Exam-style IB Maths AI HL advanced modelling questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 21 questions
- Paper 1: 20
- Paper 2: 1
- 7 easy
- 6 medium
- 5 hard
- 3 starter
- 3 worked examples
Practise Advanced Modelling questions →
AI HL formula booklet
What's examined in AI HL advanced modelling
The question bank covers these advanced modelling question types (number of questions in brackets):
- Exponential Logistic Growth (10)
- Sinusoidal Modelling (8)
- Model Fitting and Analysis (3)
Advanced Modelling worked examples
Worked example 1: Identifying Parameters of a Sinusoidal Model · easy
The depth of water in a harbour is modelled by the function $D(t) = 4\sin\left(\frac{\pi}{6}t\right) + 6$, where $D$ is the depth in metres and $t$ is the time in hours after midnight. Find the amplitude, the equation of the principal axis, and the maximum depth of the water.
1. Identify the amplitude from the coefficient of the sine function: $a = 4$.
2. Identify the principal axis from the vertical shift: $y = 6$.
3. Calculate the maximum depth by adding the amplitude to the principal axis: $6 + 4$.
4. The maximum depth is $10\text{ m}$.
Examiner tip: When extracting parameters from trigonometric models, remember that the amplitude is always a positive distance. The maximum value is simply the principal axis plus the amplitude.
Worked example 2: Solving a Logistic Population Model · medium
The population of fish in a newly stocked lake is modelled by the logistic function $P(t) = \frac{1200}{1 + 19e^{-0.25t}}$, where $t$ is the time in months. Calculate the initial population of fish, and use your Graphic Display Calculator to find the time it takes for the population to reach $600$.
1. Substitute $t = 0$ into the model to find the initial population: $P(0) = \frac{1200}{1 + 19e^0}$.
2. Evaluate the fraction: $P(0) = \frac{1200}{1 + 19} = \frac{1200}{20} = 60$. The initial population is $60$ fish.
3. Set the population equal to $600$: $600 = \frac{1200}{1 + 19e^{-0.25t}}$.
4. Graph $Y_1 = \frac{1200}{1 + 19e^{-0.25x}}$ and $Y_2 = 600$ on your GDC.
5. Find the intersection using the GDC tool (G-Solv $\rightarrow$ ISCT).
6. The time taken is $11.777\dots$, which rounds to $11.8$ months.
Examiner tip: For logistic models $y = \frac{L}{1 + Ce^{-kt}}$, the carrying capacity (maximum sustainable population) is given by the numerator $L$. In this case, the maximum population the lake can support is 1200.
Worked example 3: Solving Trigonometric Inequalities in Context · hard
Using a tidal model $D(t) = 4\cos\left(\frac{\pi}{6}t\right) + 6$, find the exact total amount of time during the first 12 hours ($0 \le t \le 12$) when the water depth is strictly greater than $8$ metres.
1. Set up the inequality: $4\cos\left(\frac{\pi}{6}t\right) + 6 > 8$.
2. Isolate the trigonometric term: $4\cos\left(\frac{\pi}{6}t\right) > 2 \implies \cos\left(\frac{\pi}{6}t\right) > 0.5$.
3. Solve the corresponding equation $\cos\left(\frac{\pi}{6}t\right) = 0.5$ using your GDC or unit circle knowledge. The principal angles are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.
4. Calculate the $t$-values: $\frac{\pi}{6}t = \frac{\pi}{3} \implies t = 2$, and $\frac{\pi}{6}t = \frac{5\pi}{3} \implies t = 10$.
5. Determine the intervals where the cosine curve is above $0.5$ within the $12$-hour domain: $0 \le t < 2$ and $10 < t \le 12$.
6. Calculate the total duration: $(2 - 0) + (12 - 10) = 2 + 2 = 4$. The water is deeper than $8\text{ m}$ for $4$ hours.
Examiner tip: When dealing with periodic inequalities, always sketch the graph on your GDC. This visual confirmation prevents you from accidentally calculating the time the water is *below* $8\text{ m}$ (which would be $8$ hours).
Try these IB Maths AI HL advanced modelling questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
A radioactive substance follows the decay model \(M(t) = 200e^{-0.1t}\). Find the initial mass and the mass remaining after exactly 10 years.
Question 2 · medium · 5 marks · Paper 1
For the logistic model \(P(t) = \frac{5000}{1 + 24e^{-0.3t}}\), find the initial population at \(t=0\).
Question 3 · hard · 7 marks · Paper 1
The population of a species of insects in a closed environment is modelled by the logistic equation:
\[P(t) = \frac{L}{1 + Ce^{-kt}}\]
Initially, the population is 20. After 5 days, the population is 80. The environment can sustain a maximum of 500 insects.
Set up the necessary equations and algebraically solve for the exact values of \(L\), \(C\), and \(k\).
All 21 advanced modelling questions with mark schemes →
FAQ
How many IB Maths AI HL advanced modelling questions are there?
There are 21 exam-style advanced modelling questions in the AI HL question bank (Paper 1: 20 · Paper 2: 1), graded 7 easy, 6 medium, 5 hard, 3 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is advanced modelling on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 20 · Paper 2: 1. Practise with your GDC — AI papers expect calculator methods throughout.
Where can I get the mark schemes?
Open the AI HL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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