Paper 3: extended problem solving

What the two long investigative questions on HL Paper 3 look like, how they move from cases to conjecture to proof, and how to keep earning marks when a part is out of reach.

IB Mathematics: Analysis and Approaches
Exam skills
HL only

11 min readLast reviewed 14 Oct 2026

Paper 3 is sat by HL students only. It consists of two long, compulsory questions, each worth roughly half of the paper's marks, and a GDC is allowed throughout. Each question is a guided investigation: it introduces a situation, asks for some concrete cases to be worked out, leads towards a pattern or general result, and then asks for that result to be proved, extended or applied. The paper rewards the habits of a mathematician at work: exploring, conjecturing, generalising and proving.

The mathematics used is the content of the course, at SL and HL, often from several topics in one question. A question may introduce a new idea or a new piece of notation, define it fully, and then ask for it to be used. Nothing outside the course is assumed, but the setting is usually unfamiliar.

How a Paper 3 question is built

A Paper 3 question is long, often a page or more of question text with many parts, but it is carefully scaffolded. The parts follow a predictable arc.

Stage	What the parts ask	Typical command terms
Getting started	Compute simple cases, fill in a table, draw a graph for a particular value	write down, find, calculate, sketch
Spotting a pattern	Describe what the cases have in common; suggest a general formula	suggest, describe, predict
Proving	Prove the conjecture, often by induction or by algebra in general	show that, prove
Extending	Apply the result to a new case, generalise to another parameter, or find a limit	hence, deduce, determine
Reflecting	Comment on a model, explain why a method fails, or justify a restriction	explain, justify, comment

The first parts are routine and carry a fair share of the marks. Almost every student can earn them, so they should never be skipped, even when the question looks unfamiliar.

Reading the question as a whole

Use the reading time to read both questions from the first line to the last. Paper 3 questions tell a story, and seeing where it ends makes the early parts easier to understand. A final part that says "prove that the area is always less than $\frac{1}{2}$ " shows what all the earlier parts are building towards.

While reading, look for:

definitions and notation introduced by the question, which must be used exactly as defined
printed results ("show that…", "given that…"), which can be used later even if they cannot be proved
parts that look independent of earlier work, which are safe places to earn marks if a middle part is difficult
the general case, usually a parameter such as $n$ or $k$ , which the question is leading towards.

Exploring with special cases

The early parts of a Paper 3 question ask for particular cases: $n = 1, 2, 3, 4$ , or a specific value of a parameter. Work these out carefully and lay them out so a pattern can be seen, usually in a table. Leave fractions unsimplified if that shows the pattern more clearly, and use the GDC for any heavy arithmetic.

Let $S_{n} = \sum_{r = 1}^{n} \frac{1}{r (r + 1)}$ , for $n \in Z^{+}$ .

(a) Find $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ .

(b) Hence suggest an expression for $S_{n}$ in terms of $n$ .

(d) Hence find the limit of $S_{n}$ as $n \to \infty$ .

Show the solution

(a) Add one term at a time.

S_{1} = \frac{1}{2}, S_{2} = \frac{1}{2} + \frac{1}{6} = \frac{2}{3}, S_{3} = \frac{2}{3} + \frac{1}{12} = \frac{3}{4}, S_{4} = \frac{3}{4} + \frac{1}{20} = \frac{4}{5}

(b) The numerator matches $n$ and the denominator is one more.

S_{n} = \frac{n}{n + 1}

When $n = 1$ : $S_{1} = \frac{1}{2}$ and $\frac{1}{1 + 1} = \frac{1}{2}$ , so $P (1)$ is true.

Assume $P (k)$ is true for some $k \in Z^{+}$ , that is, $S_{k} = \frac{k}{k + 1}$ . Then

S_{k + 1} = S_{k} + \frac{1}{(k + 1) (k + 2)} = \frac{k}{k + 1} + \frac{1}{(k + 1) (k + 2)} = \frac{k (k + 2) + 1}{(k + 1) (k + 2)}

= \frac{k^{2} + 2 k + 1}{(k + 1) (k + 2)} = \frac{(k + 1)^{2}}{(k + 1) (k + 2)} = \frac{k + 1}{k + 2}

This is $P (k + 1)$ . Since $P (1)$ is true and $P (k)$ true implies $P (k + 1)$ true, $P (n)$ is true for all $n \in Z^{+}$ by the principle of mathematical induction.

(d) Divide the numerator and denominator by $n$ .

S_{n} = \frac{1}{1 + \frac{1}{n}} \to 1 as n \to \infty

This short question follows the full arc: cases, conjecture, proof, consequence. Real Paper 3 questions are longer and less obvious, but the shape is the same.

Conjecture and proof

A conjecture is a statement believed to be true on the evidence of cases, but not yet proved. Paper 3 often asks for one and then asks for a proof, which may use any method from the course.

Choosing a method of proof

Induction suits statements about every positive integer $n$ , especially sums, products, derivatives of order $n$ and divisibility (see 1.15).
Direct algebra suits statements about a general parameter $k$ in a function, where the proof is a calculation done with letters instead of numbers.
Contradiction suits statements of impossibility: that something is irrational, or that no value of a parameter works.
A counterexample is enough to show that a general statement is false.

Writing an induction proof that earns full marks

Induction proofs appear frequently on Paper 3, and their marks are given for structure as much as for algebra. A complete proof has four parts, each written out:

The base case. Show the statement is true for the first value, evaluating both sides separately.
The assumption. "Assume the statement is true for $n = k$ , for some $k \in Z^{+}$ ", with the statement written out for $k$ .
The inductive step. Starting from one side of the statement for $n = k + 1$ , use the assumption to reach the other side. Show every line of algebra.
The conclusion. A sentence that refers to both the base case and the inductive step.

Generalising with a parameter

The other common shape of a Paper 3 question studies a family of functions or curves that depend on a parameter. The first parts use one or two particular values; later parts ask what happens for every value of the parameter.

For $k > 0$ , let $f_{k} (x) = x e^{- k x}$ .

(a) Use your GDC to find the coordinates of the maximum point of the graph of $f_{1}$ and of $f_{2}$ .

(b) Show that the maximum point of the graph of $f_{k}$ is $(\frac{1}{k}, \frac{1}{k e})$ .

(c) Hence show that the maximum points of all the graphs $y = f_{k} (x)$ lie on a single straight line, and write down its equation.

Show the solution

(a) The GDC gives the maximum of $y = x e^{- x}$ at $(1, 0.368)$ and of $y = x e^{- 2 x}$ at $(0.5, 0.184)$ .

(b) Differentiate with the product rule.

f_{k}^{'} (x) = e^{- k x} - k x e^{- k x} = e^{- k x} (1 - k x)

Since $e^{- k x} > 0$ , $f_{k}^{'} (x) = 0$ only when $x = \frac{1}{k}$ . The derivative is positive for $x < \frac{1}{k}$ and negative for $x > \frac{1}{k}$ , so this is a maximum.

f_{k} (\frac{1}{k}) = \frac{1}{k} e^{- 1} = \frac{1}{k e}

So the maximum point is $(\frac{1}{k}, \frac{1}{k e})$ , as given.

(c) At the maximum, $x = \frac{1}{k}$ and $y = \frac{1}{k e} = \frac{1}{e} \times \frac{1}{k} = \frac{x}{e}$ . This relationship does not involve $k$ , so every maximum point lies on the line

y = \frac{x}{e}

k = 0.5
k = 1
k = 2
y = x/e

As k changes, the maximum point moves, but it always stays on the line y = x/e.

The method in part (c), eliminating the parameter between the expressions for $x$ and $y$ , is a standard Paper 3 technique. It finds the curve (here a line) on which a family of special points lies. The same idea works for points of inflexion, intercepts and vertices.

When a part is out of reach

Paper 3 questions are designed so that very few students finish every part. The difference between a good and a weak score is usually how many marks are collected after the first difficulty, not whether the difficulty is overcome.

Keep going with given results

When a part says "show that" and prints the result, the result can be used in every later part whether or not it has been proved. Write "Using the result of part (b)…" and carry on.

Attempt the later parts

Later parts are often independent of the middle of the question, or depend only on a printed result. A part near the end that asks for a specific case or a comment can be quicker than the proof before it. Read every remaining part before deciding to leave a question.

Show partial progress

A partial proof earns marks for each correct step: the base case of an induction, a correct derivative, a correct substitution of the general term. Write down what is known and the first steps of an approach, even when the end is not in sight.

Communicating on Paper 3

Paper 3 answers are long, so organisation matters. Examiners need to follow the argument, and many marks are reasoning marks (R marks) for explanations, not results.

Use the notation the question defines, exactly. If the question calls an area $A_{n}$ , do not rename it $S$ .
Write sentences to link steps when the reasoning is not a chain of equations: "Since $e^{- k x} > 0$ for all $x$ , the derivative is zero only when…".
Show tables of cases neatly, with the values that show the pattern.
When the GDC is used to explore, say what was done: "Using a GDC to graph $y = f_{3} (x)$ , the maximum is at $(0.333, 0.123)$ ."
For "explain" and "justify" parts, give a mathematical reason tied to the question, not a general statement.

Preparing for Paper 3

Paper 3 tests habits more than particular content, and those habits are built by practice.

Work through past and specimen Paper 3 questions under timed conditions, a whole question in about 30 minutes.
When studying any topic, ask "what happens in general?": replace numbers with a parameter and repeat the work.
Practise induction proofs of every kind in the course: sums, derivatives, divisibility and inequalities.
Practise eliminating a parameter, finding limits of sequences of results, and moving between graphs, tables and formulas.
After each practice question, mark it strictly and note which parts were lost through communication (a missing conclusion, an undefined letter) rather than mathematics.

Checklist

To do:
I read both questions to the end during the reading time and note the printed results.
To do:
I work out early cases carefully and lay them out in a table to show the pattern.
To do:
I state conjectures precisely, with the domain of the variable.
To do:
I can write a complete induction proof with base case, assumption, inductive step and conclusion.
To do:
I can eliminate a parameter to find the curve on which a family of points lies.
To do:
I use printed results in later parts even when I could not prove them.
To do:
I split the time roughly equally between the two questions and move on when a part stalls.

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