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Paper 3 is the final exam for IB Mathematics Applications and Interpretation (AI) HL students, marking the last challenge in your IB Math AI assessment. In this post, we’ll provide a thorough breakdown of IB Math AI Paper 3, ensuring you are familiarised with the structure of the exam and know what type of questions to expect.
Math AI Paper 3
- Accounts for 20% of the student's subject grade.
- Students have 1 hour to complete Paper 3.
- Students can score up to 55 marks.
- Paper 3 is made up of 2 questions. Students must answer both questions.
- A GDC calculator is required for this paper.
- Paper 3 focuses on the HL content of the Math AI syllabus.
A typical Math AI Paper 3 question may look the following way:
Alessia is an ecologist working for Mediterranean fishing authorities. She is interested in whether the mackerel population density is likely to fall below 5000 mackerel per km3, as this is the minimum value required for sustainable fishing. She believes that the primary factor affecting the mackerel population is the interaction of mackerel with sharks, their main predator.
The population densities of mackerel (M thousands per km3) and sharks (S per km3) in the Mediterranean Sea are modelled by the coupled differential equations:
dM/dt = αM − βMS
dS/dt = γMS − δS
where t is measured in years, and α, β, γ and δ are parameters.
This model assumes that no other factors affect the mackerel or shark population densities.
The term αM models the population growth rate of the mackerel in the absence of sharks. The term βMS models the death rate of the mackerel due to being eaten by sharks.
a) Suggest similar interpretations for the following terms:
(i) γMS [1]
(ii) δS [1]
b) An equilibrium point is a set of values of M and S, such that dM/dt = 0 and dS/dt = 0.
Given that both species are present at the equilibrium point,
(i) show that, at the equilibrium point, the value of the mackerel population density is δ/y. [3]
(ii) find the value of the shark population density at the equilibrium point. [2]
c) The equilibrium point found in part (b) gives the average values of M and S over time.
Use the model to predict how the following events would affect the average value of M. Justify your answers.
(i) Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of γ is halved. No other parameter changes. [2]
(ii) Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of α is doubled. No other parameter changes. [2]
d) To estimate the value of α, Alessia considers a situation where there are no sharks and the initial mackerel population density is M0.
(i) Write down the differential equation for M that models this situation. [1]
(ii) Show that the expression for the mackerel population density after t years is M = M0eαt. [4]
(iii) Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that α = 0.549 to three significant figures. [3]
Based on additional observations, it is believed that α = 0.549, β = 0.236, γ = 0.244, δ = 1.39.
Alessia decides to use Euler’s method to estimate future mackerel and shark population densities. The initial population densities are estimated to be M0 = 5.7 and S0 = 2. She uses a step length of 0.1 years.
e) (i) Write down expressions for Mn+1 and Sn+1 in terms of Mn and Sn. [3]
(ii) Use Euler’s method to find an estimate for the mackerel population density after one year. [2]
f) Alessia will use her model to estimate whether the mackerel population density is likely to fall below the minimum value required for sustainable fishing, 5000 per km3, during the first nine years.
(i) Use Euler’s method to sketch the trajectory of the phase portrait, for 4 ≤ M ≤ 7 and 1.5 ≤ S ≤ 3, over the first nine years. [3]
(ii) Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years. [2]
(iii) State two reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid. [2]
How to succeed in Math AI Paper 3?
One of the best ways to prepare for IB Math AI Paper 3 is by practicing lots of past papers. This helps you get familiar with the question style and improves your ability to apply mathematical techniques effectively. It’s also important to do this under exam conditions so you can manage your time well and get comfortable with the format.
Another key skill is using your GDC efficiently. It can save you time on tricky calculations, graphing, and statistics while reducing mistakes. But since it’s not always easy to use, you should spend plenty of time practicing with it. The more comfortable you are with its functions, the more confident you’ll be during the exam.
We hope you found this post helpful. For more useful materials associated with the IB check out the wide variety of IA, EE and TOK exemplars available at Clastify.
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