IB Maths AI SL · Unit 2: Functions
IB Maths AI SL Linear Modelling Questions
Exam-style IB Maths AI SL linear 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.
- 12 questions
- Paper 1: 8
- Paper 2: 4
- 5 easy
- 3 medium
- 2 very hard
- 2 starter
- 3 worked examples
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What you need to know
Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →
Bacteria doubling every hour, a virus spreading through a population, radioactive decay — SL AI's exponential models are always applied. Learn to fit y = a·b^x from two data points. Exponential and logistic growth models overview →
Linear Modelling worked examples
Worked example 1: Building a linear model from two data points · easy
A tank fills at a constant rate. At $t = 0$ min it contains $10\text{ L}$; at $t = 4$ min it contains $18\text{ L}$. Write a linear model $V(t)$.
1. Identify the points: $(0, 10)$ and $(4, 18)$.
2. Gradient: $m = \frac{18 - 10}{4 - 0} = 2$.
3. $y$-intercept: at $t = 0$, $V = 10$, so $c = 10$.
4. Formulate: $V(t) = mt + c$.
5. State: $\mathbf{V(t) = 2t + 10}$.
Examiner tip: If the data gives a value at $t = 0$, the $y$-intercept is handed to you for free — no point-slope substitution needed.
Worked example 2: Rate of straight-line depreciation · medium
Machinery bought for $\text{€}20\,000$ depreciates linearly to $\text{€}8000$ over $6$ years. Calculate the exact annual rate of depreciation.
1. Identify the coordinates: $(0, 20000)$ and $(6, 8000)$.
2. Gradient: $m = \frac{8000 - 20000}{6 - 0}$.
3. Simplify: $m = \frac{-12000}{6}$.
4. Evaluate: $m = -2000$.
5. State: annual rate of depreciation $= \mathbf{\text{€}2000\text{ per year}}$.
Examiner tip: "Depreciation" is the AMOUNT lost per year and is reported as a positive number, even though the gradient itself is negative.
Worked example 3: Comparing two linear cost models · hard
Plumber A charges $\text{€}40 + \text{€}15h$. Plumber B charges $\text{€}60 + \text{€}10h$. Find the exact number of hours for which both plumbers charge the same total.
1. Model A: $C_A(h) = 40 + 15h$.
2. Model B: $C_B(h) = 60 + 10h$.
3. Equate: $40 + 15h = 60 + 10h$.
4. Collect: $40 + 5h = 60$.
5. Solve: $5h = 20 \implies h = \mathbf{4\text{ hours}}$.
Examiner tip: Write both linear models $y = mx + c$ explicitly BEFORE equating them — that prevents confusing fixed fees with hourly rates.
Try these IB Maths AI SL linear 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 · 3 marks · Paper 1
A car rental company charges a flat daily rate of \(\text{€}40\) plus \(\text{€}0.15\) for every kilometre driven.
- Write down a linear function \(C(x)\) to model the total daily cost of renting the car and driving it for \(x\) kilometres.
- Calculate the total cost of renting the car for one day and driving exactly \(250 \text{ km}\).
Question 2 · medium · 5 marks · Paper 2
An electrician charges a fixed callout fee of \(\text{€}C\) plus an hourly rate of \(\text{€}R\). A 2-hour job costs \(\text{€}130\), and a 5-hour job costs \(\text{€}235\).
- Write down two linear equations representing this information.
- Find the values of \(C\) and \(R\).
- Calculate the total cost of a job that takes 3.5 hours.
Question 3 · very hard · 7 marks · Paper 2
A water tank is draining. The volume of water, \(V\) in litres, remaining in the tank after \(t\) minutes is modelled by \(V(t) = a t^2 + b t + c\). Initially, the tank contains 500 litres. After 10 minutes, there are 200 litres remaining. After 20 minutes, the tank is completely empty.
- Write down three linear equations to find \(a, b, c\).
- Using your GDC, find the exact values of \(a, b, c\).
- Find the rate at which the water is draining exactly 5 minutes after it started.
All 12 linear modelling questions with mark schemes →
FAQ
How many IB Maths AI SL linear modelling questions are there?
There are 12 exam-style linear modelling questions in the AI SL question bank (Paper 1: 8 · Paper 2: 4), graded 2 very hard, 3 medium, 5 easy, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is linear modelling on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 8 · Paper 2: 4. Practise with your GDC — AI papers expect calculator methods throughout.
Where can I get the mark schemes?
Open the AI SL 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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