Paper 3 guide · Analysis and approaches HL
IB Math AA HL Paper 3 in 4 steps
8 kinds of question come up again and again. Learn to spot each one and the method that earns the marks. About 45 minutes from start to finish.
- 1 hourto answer
- 55marks
- 2long questions
- 20%of your grade
- GDCallowed
Know the paperWhat Paper 3 is and the arc every question follows3 min
Paper 3 is one hour long and worth 55 marks. It has two extended questions, each worth between 23 and 32 marks, and a GDC is allowed. It is 20% of your HL grade. Paper 1 is 30%, Paper 2 is 30% and the internal assessment is 20%.
Papers 1 and 2 test topics one at a time. Paper 3 gives you one situation and walks you through it in 3 to 10 lettered parts, 7 on average. A function with a parameter, a sequence, a polynomial, a differential equation: you explore it, conjecture, then prove.
The parts depend on each other. A value from part (b) feeds part (e). In 38 of the 42 questions at least one part prints its answer, so you can use that result even if your own derivation failed. A "show that" part earns marks for the chain of steps, and a line that only restates the target earns nothing.
Notes alone do not train this. You need to sit a whole question against the clock, write full reasons and see where marks go. Every question follows the same six-stage pattern, and the eight types below cover all 42 questions we read.
Every question follows the same arc
- OrientShort opening parts: a sketch, the first few terms, the name of a test. Later parts reuse these numbers, so do them carefully.
- StructureThe derivation everything rests on, usually a show that. Write a chain from what you know to the printed line.
- ExploreThe GDC earns the facts: vary a parameter, tabulate a sequence, run a regression or a test.
- DeduceThe pattern is proved or explained. This is where the proof and reasoning marks sit.
- GeneraliseThe longest parts. They repeat an earlier argument with letters, or add a twist such as a new starting value.
- InterpretOne or two marks for a sentence: a meaning, a comparison or a limit of the model. Reach it even after a bad middle.
Learn the 8 question typesHow to spot each one, the method and where marks go15 min
We sorted all 42 questions in 21 papers, from the specimen paper to May 2026, into 8 types. Most common first. Open one at a time.
Families of curves10 of 42 questions
You are given a function with a parameter, either a real constant or an index $n$. Early parts sketch two or three members, a GDC sweep leads to a conjecture, and the last parts prove it for every member.
How to spot it
- A function with an extra letter, such as $f_a(x)$ or $f_n(x)$
- Phrases like "every member" or "for all values of $a$"
- A request to count stationary points or roots as the parameter changes
The method
- Sketch two or three members and label what they share.
- Find what every member has in common by setting the coefficient of the parameter to zero.
- Differentiate, then ask when $f'(x)=0$ has solutions. That decides the count of features.
- Tabulate on the GDC for several parameter values and state a conjecture with its condition.
- Prove both directions of any "if and only if", and every case you claimed.
- To find the curve a feature lies on, solve for the parameter and substitute.
Where marks are lost
- A conjecture with no condition on the parameter.
- Proving one direction of an "if and only if" and stopping.
- Missing a case: a negative parameter, or odd and even $n$.
On your GDC. Graph several members from a table of parameter values and count stationary points or roots for each. This builds the conjecture before you prove it.
Differential equations6 of 42 questions
A first-order equation, often inside a short modelling story, is solved with a starting value. Later parts use Euler's method, describe the family of solution curves or ask what a long-term value means.
How to spot it
- A derivative given in terms of both $x$ and $y$
- The phrase "rate of change is proportional to"
- A request for a slope field, a family of solutions or a step size $h$
The method
- Name the kind first: separable, homogeneous (substitute $y=vx$) or linear (integrating factor).
- Carry out that method in full, with an integral written on each side.
- Add the constant at once and use the starting value to fix it.
- For Euler's method write the formula once, then build the table row by row.
- Describe the family: where solutions rise or fall, equilibria, asymptotes.
- Say what a long-term value means in the story.
Where marks are lost
- Substituting the starting value before the constant of integration has been added.
- Stopping at an implicit form when "$y=$" is asked for.
- Rounding in early Euler steps so the final value drifts.
On your GDC. The GDC builds Euler tables and graphs solution curves. It does not solve the equation for you, so use it to check the formula you derived.
Polynomials and complex roots6 of 42 questions
A polynomial, or an equation such as $z^n=w$, has roots with a pattern. You relate roots to coefficients, sum powers of the roots and finish with a condition that holds for every $n$.
How to spot it
- Roots described as a geometric or arithmetic sequence
- Roots of $z^n=w$ or the roots of unity
- A sum of powers of the roots
The method
- Write the polynomial two ways: with its roots and with its coefficients.
- Match coefficients to get the sum of the roots, the sum of products in pairs and the product.
- For $z^n=w$ find the modulus and argument, then space the roots by $\frac{2\pi}{n}$.
- Plot the roots on an Argand diagram and describe the shape they make.
- Prove the property for one root, then for every $n$.
- Check the general result with $n=3$ or $n=4$.
Where marks are lost
- Using conjugate pairs when the coefficients are not all real.
- Giving decimals when an exact form is asked for.
- Checking a result for one value of $n$ and claiming it for all.
On your GDC. Solve the polynomial numerically and check your root relations against the values before you write the proof.
Sequences and induction6 of 42 questions
A rule for a sequence, or a counting story, gives you the first few terms. You conjecture a formula, prove it by induction or telescoping, then use it for a limit or a bound.
How to spot it
- A recurrence such as $u_{n+1}=\dots$
- A counting story (tiles, paths, arrangements) that leads to a recurrence
- "Prove by induction" or "show that the sum telescopes"
The method
- Compute the first four to six terms exactly.
- Look at differences, ratios or a comparison with $2^n$ and $n!$.
- State the conjecture with its range of $n$.
- Prove by induction: base case, assumption, step, conclusion. Use two base cases for a two-step rule.
- For a sum, write each term as $f(k)-f(k+1)$ and cancel.
- For a limit, divide through by the dominant term.
Where marks are lost
- No base case, or no closing sentence for the induction.
- One base case for a rule that needs two.
- Using the assumption without writing "assume true for $n=k$".
On your GDC. Build a table of terms on the GDC. A tidy table of the first six terms usually shows the pattern.
Curves, surfaces and optimisation5 of 42 questions
A picture of a shape, a solid or a vessel comes first. A formula is built for one case, then in general, then optimised or differentiated with respect to time.
How to spot it
- A diagram with lengths and an angle or a depth
- "Find the maximum" or "show that this is a minimum"
- A rate of change of a volume, a depth or a length
The method
- Mark every length on the diagram and name the variables.
- Use the constraint to leave one variable.
- Differentiate the quantity and set the derivative to zero.
- Justify the maximum or minimum with a sign change or the second derivative.
- For rates, chain the derivatives: $\frac{dV}{dt}=\frac{dV}{dh}\times\frac{dh}{dt}$.
- Check the answer against the domain and units.
Where marks are lost
- An optimum with no justification that it is a maximum or a minimum.
- Differentiating before writing the quantity in one variable.
- Leaving out units or the domain.
On your GDC. Graph the quantity to be optimised. The GDC shows the turning point, which checks your calculus.
Probability models4 of 42 questions
A short story builds a probability model, such as waiting for a first success. You show a distribution or formula, use a series or calculus for an exact expectation, then let the number of trials or the probability become a letter.
How to spot it
- "Show that $\mathrm{P}(X=k)=\dots$"
- An expected number of trials, or a "least $n$"
- A probability $p$ that is allowed to vary
The method
- List the cases for a small version of the story.
- Define the random variable and state its possible values.
- Build each probability as a product of stage probabilities.
- Sum with a geometric series, or differentiate or integrate for an expectation.
- Treat probabilities as polynomials in $p$ and compare, maximise or solve.
- End with one sentence on what the model leaves out.
Where marks are lost
- Not stating that trials are independent.
- Dividing or taking logs and forgetting that the inequality can reverse.
- Summing an infinite series without saying why it converges.
On your GDC. Evaluate long sums quickly and graph a probability against $p$ to see where two models cross.
Limits and approximation3 of 42 questions
A series is built for a function, used to estimate a value or find a limit, and then examined for the size of its error.
How to spot it
- "Maclaurin series up to the term in $x^n$"
- "Find the limit as $x\to0$"
- "Find the least $n$ for which the error is less than"
The method
- Build the series from known series, or from derivatives at $0$, and show each coefficient.
- Choose an $x$ that is small and inside the interval of validity.
- For a limit, use the series in one line or l'Hôpital once for each $\frac00$ form.
- Bound the error with the first omitted term (alternating series) or a geometric bound.
- For a least $n$, solve the inequality, then test $n-1$ and $n$.
Where marks are lost
- Skipping a coefficient that the question says to show.
- Using l'Hôpital when the form is not $\frac00$ or $\frac\infty\infty$.
- Giving a least $n$ without checking the value on each side.
On your GDC. Tabulate partial sums and errors on the GDC to find the least $n$, then check it by hand.
Vectors and motion2 of 42 questions
Two objects move along straight lines, each given by a position and a velocity. You find where the lines meet or how close the objects come, and use the scalar product for an angle.
How to spot it
- A line written as $\mathbf{r}=\mathbf{a}+t\mathbf{b}$
- "Closest approach" or "minimum distance"
- "Show that the paths do not cross"
The method
- Write each position as a vector in $t$.
- Form the displacement vector between the two objects.
- Write the squared distance as a quadratic in $t$.
- Minimise it by completing the square or by calculus.
- Use the scalar product for the angle between directions.
- Give the distance with its units, and the time it happens.
Where marks are lost
- Using the same parameter for both lines when testing for an intersection.
- Finding the time of closest approach but not the distance.
- Stopping at the squared distance and forgetting the square root.
On your GDC. Graph the distance against $t$ to see the closest approach and check your algebra.
Work through one investigationWobbly lines, 26 marks, part by part25 min
Wobbly lines · 26 marks
This question asks you to investigate the family of functions $f_a(x)=x+a\sin x$, $0\le x\le2\pi$, where $a$ is a positive real constant.
Try each part yourself first, then open its solution. M is a method mark, A accuracy, R reasoning and AG a given answer.
(a) Sketch $y=f_{0.5}(x)$ and $y=f_2(x)$ together on one set of axes, labelling the coordinates of the end points. [3]
Show the solution
- Both curves start at $(0,0)$ and end at $(2\pi,2\pi)$; both cross at $(\pi,\pi)$A1End points and the crossing point labelled.
- $a=0.5$: increasing throughoutA1Shape of the first member.
- $a=2$: rises to a maximum, falls to a minimum, then rises againA1Shape of the second member.
(b(i)) Show that every member of the family passes through the points $(0,0)$, $(\pi,\pi)$ and $(2\pi,2\pi)$. [2]
Show the solution
- $\sin0=\sin\pi=\sin2\pi=0$M1The $a$-term vanishes at each of the three values.
- so $f_a(0)=0$, $f_a(\pi)=\pi$ and $f_a(2\pi)=2\pi$ for every $a$A1Common points: set the coefficient of $a$ to zero.
(b(ii)) Show that $f_a(2\pi-x)=2\pi-f_a(x)$, and describe the symmetry of the graph that this shows. [2]
Show the solution
- $f_a(2\pi-x)=2\pi-x+a\sin(2\pi-x)=2\pi-x-a\sin x=2\pi-f_a(x)$A1$\sin(2\pi-x)=-\sin x$.
- rotational symmetry of order 2 about $(\pi,\pi)$R1The midpoint of $(x,f_a(x))$ and $(2\pi-x,\,2\pi-f_a(x))$ is $(\pi,\pi)$.
(c) Show that, for every value of $a$, the only point of inflexion of the graph with $0<x<2\pi$ is at $(\pi,\pi)$. [3]
Show the solution
- $f_a'(x)=1+a\cos x$ and $f_a''(x)=-a\sin x$A1Both derivatives.
- $f_a''(x)=0$ for $0<x<2\pi$ only at $x=\pi$M1$\sin x=0$ in the interval only at $\pi$.
- $\sin x$ changes sign at $\pi$, so $f_a''$ does too: an inflexion at $(\pi,\pi)$R1The sign change is stated, not only $f''=0$.
(d) Use your GDC to find the number of stationary points with $0<x<2\pi$ for $a=0.5,\,1,\,1.5$ and $2$. Hence suggest a condition on $a$ for the graph to have two stationary points. [3]
Show the solution
- GDC, number of stationary points: $a=0.5$: 0; $a=1$: 1; $a=1.5$: 2; $a=2$: 2A1 A1Table from graphs of $y=1+a\cos x$.
- Conjecture: two stationary points when $a>1$A1A conjecture carries its condition.
(e) Prove that the graph has stationary points with $0<x<2\pi$ if and only if $a\ge1$, and find their $x$-coordinates in terms of $a$ when $a>1$. [4]
Show the solution
- $f_a'(x)=0\iff\cos x=-\dfrac1a$M1Set the derivative to zero.
- A solution needs $-1\le-\dfrac1a\le1$, that is $a\ge1$ (as $a>0$); conversely, if $a\ge1$ then $-\dfrac1a\in[-1,0)$ and a solution existsR1Both directions of the “if and only if”.
- For $a>1$: $x=\arccos\!\left(-\dfrac1a\right)$ or $x=2\pi-\arccos\!\left(-\dfrac1a\right)$A1 A1One mark for each solution in $(0,2\pi)$.
(f) Show that, for $a>1$, every stationary point lies on the curve $y=x-\tan x$. [3]
Show the solution
- At a stationary point $\cos x=-\dfrac1a$, so $a=-\dfrac1{\cos x}$M1Solve for the parameter.
- $y=x+a\sin x=x-\dfrac{\sin x}{\cos x}$A1Substitute for $a$.
- $y=x-\tan x$AGPrinted result; the line above it must be shown.
- $\cos x\ne0$ because $\cos x=-\dfrac1a$R1The division is valid.
(g) Find the exact coordinates of the local maximum point and the local minimum point of $y=f_2(x)$. [3]
Show the solution
- $a=2$: $\cos x=-\dfrac12$, so $x=\dfrac{2\pi}3$ or $\dfrac{4\pi}3$A1From part (e).
- $f_2''\!\left(\dfrac{2\pi}3\right)=-2\sin\dfrac{2\pi}3<0$: local maximum at $\left(\dfrac{2\pi}3,\ \dfrac{2\pi}3+\sqrt3\right)$A1Exact value, no decimals.
- local minimum at $\left(\dfrac{4\pi}3,\ \dfrac{4\pi}3-\sqrt3\right)$A1$f_2''>0$ there, or use the symmetry of (b)(ii).
(h) Find the greatest value of $a$ for which the graph of $y=f_a(x)$ is contained in the square $0\le x\le2\pi$, $0\le y\le2\pi$. [3]
Show the solution
- Local maximum value $M(a)=\arccos\!\left(-\dfrac1a\right)+\sqrt{a^2-1}$M1$\sin(\arccos u)=\sqrt{1-u^2}$, then $a\sqrt{1-1/a^2}=\sqrt{a^2-1}$.
- Solve $M(a)=2\pi$: $a=4.60$A1GDC (stored $4.60334$). Do not round $M$ first.
- $M$ increases with $a$; by (b)(ii) the local minimum is $2\pi-M(a)$, so it stays above $0$ exactly when $M(a)\le2\pi$R1Why this is the greatest value, not just a value.
Plan exam dayThe hour minute by minute, the mistakes to avoid, the weeks before5 min
How to spend the hour
- 0–3 minRead both questions. Note the printed results in each and which question looks friendlier. Start with the one whose first parts you can do.
- 3–8 minFirst question, opening parts: sketches, first terms, values. These are quick marks that later parts reuse, so write them cleanly.
- 8–18 minFirst question, structure and explore: the show-that, then the GDC table or sweep. Write what you entered and the values you read.
- 18–30 minFirst question, deduce and generalise: the conjecture and its proof, then the longest parts. Stop at 30 minutes even if a part is unfinished.
- 30–35 minSecond question, opening parts, with the same care as the first.
- 35–45 minSecond question, structure and explore.
- 45–55 minSecond question, deduce and generalise. If you are stuck, use a printed result and move on.
- 55–60 minGo back to the gaps. Check each show-that chain, the conditions in your conjectures, exact values and the last interpret part.
Five ways marks are lost
- 1Jumping to the printed answer. A show-that part is marked on the route. Start from what you know, write every line and end at the printed result. Never work backwards from it.
- 2A conjecture with no condition. Write the rule and the range it holds for, such as "two stationary points when $a>1$". Then prove both directions of any "if and only if".
- 3Incomplete induction. Write the base case, the assumption for $n=k$, the step and the closing sentence. Each is a separate mark, and the closing sentence is the one students drop.
- 4Rounding early or giving decimals. Keep full GDC precision in your working and give exact values when the question asks. Round only the final answer, to 3 significant figures.
- 5Spending too long on the early parts. Note the time you started each question. At 30 minutes, move on. The later parts carry more marks, and the first parts are quick if you do not polish them.
The weeks before
- Wk 1–2Repair the toolkit Paper 3 draws on: calculus, complex numbers, induction, series, differential equations, vectors and probability. Use short exercises, not whole papers.
- Wk 3Learn the six stages and the mark codes. Read one worked investigation slowly, cover the solution and compare line by line.
- Wk 4–5Work the three most frequent types in full: families of curves, differential equations, then polynomials and complex roots or sequences. One question per sitting, untimed, writing everything.
- Wk 6Cover the remaining types with one question each. Keep an error log of lost marks by type and stage.
- Wk 7Sit full Paper 3s in one hour, with a GDC and no notes. Mark each against a written markscheme and rewrite the parts you lost.
- Wk 8Redo your weakest type. Reread the error log, the mark codes and the hour plan. Sit one last paper two or three days before the exam, then stop.
IB Math AA HL Paper 3 questions
Is Paper 3 hard?
It is the hardest paper to start, because the context is unfamiliar and the parts are linked. It is also the most predictable. The 42 questions we read fall into eight types, and 38 of them print an answer in at least one part, so a stuck step does not end the question.
How long is Paper 3 and what is it worth?
Paper 3 is 1 hour long and worth 55 marks, with two extended questions. It counts for 20% of your HL grade. Papers 1 and 2 count for 30% each and the internal assessment for 20%.
How do I practise with few past papers?
Take each question in stages. Do the early parts timed, cover the later ones, then write your own conjecture before looking. Rework a question with a changed parameter, such as $a\sin x$ replaced by $a\cos x$. Our analysis gives you the eight types, so you can build practice by type.
Can I use my GDC?
Yes. Paper 3 allows a GDC and expects you to use it for tables, graphs and numerical solving. Write what you entered and the value you read. Many marks are for exact algebra or a proof, which no calculator can write for you.
What topics come up most?
Families of curves come up most: 10 of the 42 questions and 271 of the 1155 marks, with at least one in every year of the analysis. Differential equations, polynomials and complex roots, and sequences and induction follow with 6 questions each. Probability models appear only from 2024.
How is it marked?
Marks are given for method (M), accurate results (A) and reasons (R). Follow-through lets a correct later step score after an earlier slip. On a show-that part every line of the route is marked and a line that only restates the target earns nothing.
Every type in its own chapter, worked investigations with every mark explained, and six full practice papers.
Past-paper analysis · Predicted papers · Practice by topic · IB Math AI HL Paper 3