IB Maths AI SL · Unit 2: Functions
IB Maths AI SL Direct/Inverse Variation and Cubic Models Questions
Exam-style IB Maths AI SL direct/inverse variation and cubic models 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.
- 10 questions
- Paper 1: 7
- Paper 2: 3
- 2 easy
- 4 medium
- 2 hard
- 2 very hard
- 3 worked examples
Practise Direct/Inverse Variation and Cubic Models questions →
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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 →
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What's examined in AI SL direct/inverse variation and cubic models
The question bank covers these direct/inverse variation and cubic models question types (number of questions in brackets):
- Direct/Inverse Variation (5)
- Properties of Cubic Functions (3)
- Constructing Cubic Models (2)
Direct/Inverse Variation and Cubic Models worked examples
Worked example 1: Direct variation with cubes · easy
$y$ varies directly as the cube of $x$. When $x = 2$, $y = 40$. Find the constant of proportionality $k$, and hence $y$ when $x = 3$.
1. Model: $y = kx^3$.
2. Substitute: $40 = k(2)^3$.
3. Solve: $40 = 8k \implies k = 5$.
4. State the specific model: $y = 5x^3$.
5. Substitute $x = 3$: $y = 5(3)^3 = 5(27) = \mathbf{135}$.
Examiner tip: Direct variation always has the form $y = kx^n$; inverse variation always $y = \frac{k}{x^n}$. Identify which one first, THEN substitute your initial condition.
Worked example 2: Inverse square variation — light intensity · medium
Light intensity $I$ varies inversely as the square of distance $d$. When $d = 4\text{ m}$, $I = 15\text{ units}$. Find the distance at which the intensity drops to $2.4\text{ units}$.
1. Model: $I = \frac{k}{d^2}$.
2. Substitute $(d = 4, I = 15)$: $15 = \frac{k}{16}$.
3. Solve for $k$: $k = 240$. Model: $I = \frac{240}{d^2}$.
4. Substitute $I = 2.4$: $2.4 = \frac{240}{d^2} \implies d^2 = 100$.
5. Take the positive root: $d = \mathbf{10\text{ m}}$.
Examiner tip: When the problem says "inverse variation with the square (or cube) of", the exponent on the denominator must match. Forgetting the exponent is the classic error here.
Worked example 3: Maximising a cubic volume model · hard
An open box is made from a $30\text{ cm} \times 30\text{ cm}$ square by cutting squares of side $x\text{ cm}$ from each corner and folding the sides up. Its volume is $V(x) = x(30 - 2x)^2$. Using your GDC, find the $x$ that maximises the volume and the maximum volume.
1. Enter $Y_1 = x(30 - 2x)^2$ on the GDC.
2. Set a valid domain: $30 - 2x > 0 \implies 0 < x < 15$.
3. Use the maximum tool (G-Solv $\to$ MAX).
4. Read: vertex $(5, 2000)$.
5. State: $\mathbf{x = 5\text{ cm}}$, max volume $\mathbf{2000\text{ cm}^3}$.
Examiner tip: In real-world modelling, always set the practical domain (here $0 < x < 15$) before running the GDC max tool — otherwise you might accept a mathematical maximum that has no physical meaning.
Try these IB Maths AI SL direct/inverse variation and cubic models 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
The variable \(y\) varies directly with the cube of \(x\). When \(x = 3\), \(y = 54\).
- Find the constant of proportionality, \(k\), and write down the equation linking \(y\) and \(x\).
- Calculate the value of \(y\) when \(x = 5\).
Question 2 · medium · 4 marks · Paper 1
The force of attraction, \(F\) Newtons, between two magnets varies inversely with the square of the distance, \(d\) cm, between them. When \(d = 4\), \(F = 150\).
- Find the equation linking \(F\) and \(d\).
- Calculate the force when the distance is increased to \(10 \text{ cm}\).
Question 3 · hard · 6 marks · Paper 2
A rectangular sheet of metal measures \(30 \text{ cm}\) by \(20 \text{ cm}\). Four identical squares of side length \(x \text{ cm}\) are cut from the corners. The sides are then folded up to form an open rectangular box.
- Show that the volume of the box is modelled by the cubic function \(V(x) = 4x^3 - 100x^2 + 600x\).
- State a suitable domain for \(x\) in this context.
- Using your Graphic Display Calculator, find the value of \(x\) that maximizes the volume, and state the maximum volume.
All 10 direct/inverse variation and cubic models questions with mark schemes →
FAQ
How many IB Maths AI SL direct/inverse variation and cubic models questions are there?
There are 10 exam-style direct/inverse variation and cubic models questions in the AI SL question bank (Paper 1: 7 · Paper 2: 3), graded 2 easy, 4 medium, 2 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is direct/inverse variation and cubic models on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 7 · Paper 2: 3. 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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