Paper 1: short-response questions

Plan your time on Paper 1, recognise the short question shapes that recur, and write down enough working that the GDC's answers earn every mark.

IB Mathematics: Applications and Interpretation
Exam skills
SL and HL

20 min readLast reviewed 15 Oct 2026

Introduction

Paper 1 is a set of compulsory short-response questions spread across the whole course. Each question has one context and between one and five parts, and the paper tests breadth: a typical session moves from a sequence to a triangle, a hypothesis test, a derivative and a loan in the space of an hour and a half. Because the questions are short, the marks are lost in small ways: a misread part, a missing line of working, an answer rounded too early or left in calculator notation.

A graphic display calculator is allowed on the whole paper, as it is on every Applications and interpretation paper. That changes what the paper rewards. Most of the arithmetic is done by the calculator, so the marks go to choosing the right method, setting it up correctly, writing down what was entered and giving the answer in the right form. The skills that follow apply to Paper 1 at both levels; HL Paper 1 is longer and its later questions use HL topics, but the shapes and the habits are the same.

The overview of the assessment gives the duration and total marks of each paper. On Paper 1 those numbers set a pace of a little over a minute per mark, and that pace shapes almost every decision in the paper.

How a Paper 1 question is built

A Paper 1 question starts with a short stem: a sentence or two of context, sometimes a diagram or table, and often a function or model. The parts that follow are labelled (a), (b), (c) and so on, each with its marks in square brackets. In the five SL Paper 1s set in 2021 and 2022, every paper had 13 questions, and the totals per question ran from 4 to 9 marks, with 6 the most common value. Individual parts are worth 1 to 5 marks.

Each part opens with a command term, a word such as "write down", "find" or "show that" that tells you what kind of answer is wanted and how much working to show. Recognising the command term before reading the rest of the sentence is the single most useful reading habit on Paper 1, because it tells you whether the answer is a single extracted value or a method with stages.

The full list of command terms, with what each one demands, is on the "Command terms" page of Exam skills. On Paper 1 four of them do most of the work: "write down" (usually 1 mark, no working needed), "find" and "calculate" (a method with stages), and "show that" (a given result reached by working forwards).

The shapes that recur

Paper 1 questions fall into a small number of shapes. Knowing the shape on the first read lets you plan the whole question before writing anything.

Shape	What the parts ask	Typical marks
A model given as a function	Write down a parameter or the initial value; find a maximum, a zero or an intersection on the GDC; interpret the answer in context	5–7
A sequence in context	Write down the first term and the common difference or ratio; show that a term has a given value; find a sum or the first term to pass a target	5–7
A hypothesis test from a table	Write down the hypotheses and the degrees of freedom; show that an expected frequency is correct; write down the p-value; state a conclusion with a reason	6–8
Finance with the TVM solver	Find a final value, a payment or a number of periods; give money to the stated accuracy	5–7
A triangle or a solid	Find a side or angle with the sine or cosine rule; find an area or a volume; give units	5–7
Differentiation	Differentiate; find the gradient at a point; find the equation of a tangent or normal in a stated form	5–7
A probability setting	Complete a tree diagram or a table; find a probability; find an expected number	4–7

The questions are not all the same difficulty. Questions near the start of the paper tend to be routine, and the last three or four questions usually contain at least one part that needs an idea rather than a technique. Within a question, the first part is almost always the easiest, often a 1-mark "write down".

Which topics appear most

Questions come from all five topics of the course, though not every subtopic appears in every session. The table counts the 65 questions on the five SL Paper 1s of 2021 and 2022 by the main topic of each question, next to each topic's share of SL teaching time.

Topic	Questions	Share of questions	Share of SL teaching time
1 Number and algebra	8	12%	13%
2 Functions	15	23%	26%
3 Geometry and trigonometry	15	23%	15%
4 Statistics and probability	17	26%	30%
5 Calculus	10	15%	16%

Source: IB questionbank, Mathematics: applications and interpretation SL Paper 1, May 2021 to May 2022 (five papers); each question counted once by its main topic.

The shares of questions follow the teaching time closely, with geometry and trigonometry slightly over-represented because triangles, solids and Voronoi diagrams make natural short questions. Statistics and probability is the largest topic on the paper. Within it, hypothesis testing (subtopic 4.11) and probability (4.5 and 4.6) appear most often, followed by statistical diagrams (4.2) and the normal distribution (4.9). Many questions also combine two topics, for example a function model whose maximum is found by differentiation, or a geometric sequence set in a finance context.

Time management: about a mark a minute

SL Paper 1 has 80 marks in 90 minutes, which is 1.125 minutes per mark. HL Paper 1 has 110 marks in 120 minutes, about 1.09 minutes per mark. The working rule at both levels is about a mark a minute, with the spare time saved for checking and for the harder final questions.

SL: \frac{90}{80} = 1.125 min per mark HL: \frac{120}{110} \approx 1.09 min per mark

The marks on a question are therefore a time budget. A 6-mark question deserves about 6 or 7 minutes. If it has taken 10 minutes and the answer is still not in sight, the best move is to write down whatever method you have, leave a gap and come back at the end.

Marks on the question	Time budget (SL)	Time budget (HL)
4	4 min 30 s	4 min 22 s
6	6 min 45 s	6 min 33 s
8	9 min	8 min 44 s
80 (whole SL paper)	90 min	–
110 (whole HL paper)	–	120 min

The marks on a part also show how much work it needs. A 1-mark part is answered in one line. A 2-mark part is usually one method step and an answer, often marked (M1)A1. A 4- or 5-mark part has several stages, and the marks are spread across them. If the answer to a 1-mark part needs half a page, the question has probably been misread.

SL Paper 1 has 13 questions. After 50 minutes a candidate has completed questions 1 to 8, worth 46 marks in total. The remaining questions are worth 34 marks. Determine whether the candidate is on schedule, and suggest how the remaining time should be used.

Show the solution

At about 1.125 minutes per mark, 46 marks should take about

46 \times 1.125 = 51.75 minutes

so the candidate is slightly ahead of schedule, by about 2 minutes.

The remaining 34 marks need about $34 \times 1.125 = 38.25$ minutes, and there are $90 - 50 = 40$ minutes left. That leaves under 2 minutes of spare time, so there is no room for a long check at the end. The candidate should work through questions 9 to 13 in order, skip any part that stalls for more than a minute or two beyond its budget, and use any time left for those skipped parts first, before checking finished answers.

Order and skipping

Questions are compulsory, so every one must be attempted, but they need not be answered in order. A common plan is one steady pass through the paper in order, answering everything that goes smoothly and marking anything left unfinished, followed by a second pass on the marked parts. Skipping is especially useful for "show that" parts: if a "show that" part cannot be done, the given result can still be used in the next part, so the rest of the question is not lost.

What to write down when the GDC does the work

Since the calculator finds most numerical answers, it is tempting to write only the answer. That is safe only when the answer is right. Markschemes give two kinds of method mark: implied marks, printed in brackets as (M1) or (A1), which are awarded automatically when the final answer is correct, and unbracketed marks, M1 or A1, which need the working to be seen. A correct answer with no working earns the bracketed marks but none of the unbracketed ones. A wrong answer with no working earns nothing at all.

The safest habit is to write one line that shows what was given to the calculator, then the unrounded result, then the final answer. It costs a few seconds and protects marks whenever the answer is wrong. It also lets a correct method earn follow-through marks in later parts.

What the GDC did	What to write down
Solved an equation	The equation, as entered: $- 4.9 t^{2} + 29.4 t + 1.5 = 0$
Found a maximum, minimum or intersection on a graph	The function(s) graphed and the feature found; a small labelled sketch if it helps
Solved a system of equations	The equations, with variables defined
Used the finance (TVM) solver	Every entry: N, I%, PV, PMT, FV, P/Y, C/Y
Calculated a probability	The distribution with its parameters and the probability found: $X \sim B (20, 0.3)$ , $P (X \leq 4)$
Ran a hypothesis test	The test named, the statistic or p-value, the comparison with the significance level
Found a regression line	The regression type and the equation in the question's variables

The GDC skills behind each row, with the menus on common models, are on the "Using the GDC" page of Exam skills. The rules for accuracy, rounding and notation are on "Accuracy, notation and setting out".

Three candidates answer the same 2-mark part: "Find the time when the rocket hits the ground", where the height is $h (t) = - 4.9 t^{2} + 29.4 t + 1.5$ and the correct answer is $t = 6.05$ seconds. The markscheme is $- 4.9 t^{2} + 29.4 t + 1.5 = 0$ (M1), $t = 6.05$ A1. Explain how many marks each answer earns.

Candidate A writes: $t = 6.05$ s
Candidate B writes: $- 4.9 t^{2} + 29.4 t + 1.5 = 0$ , so $t = 6.15$ s
Candidate C writes: $t = 6.15$ s

Show the solution

Candidate A gives the correct answer. The method mark is bracketed, so it is implied by the correct answer: 2 marks out of 2.

Candidate B writes the correct equation, which earns the (M1) on its own, but the answer is wrong (probably a value copied incorrectly from the screen), so the A1 is lost: 1 mark out of 2.

Candidate C gives a wrong answer with no working. Nothing shows a correct method, so nothing can be implied: 0 marks out of 2.

Candidates A and C wrote the same amount of working, and the difference in their scores comes only from whether the answer happened to be right. Candidate B's single line of working earned a mark despite the slip.

Setting up is the method

On a calculator paper the method mark is usually for the set-up: the equation formed, the system written, the distribution chosen, the integral with its limits. That is where the mathematics is, and it is what the examiner needs to see. A question that says "find the number of tickets of each type sold" earns most of its marks for writing three correct equations; pressing the solver earns very little on its own.

Defining variables is part of the set-up. Writing "let $t$ be the price of a tea in dollars" before the equations takes seconds and makes every later line readable. Use the letters the question uses; if the question calls the time $t$ , do not switch to $x$ because the calculator does.

Worked Paper 1 questions

The questions that follow are written in the style of Paper 1, with markscheme-style solutions. The marks show where credit is given.

The height, $h$ metres, of a model rocket $t$ seconds after launch is modelled by $h (t) = - 4.9 t^{2} + 29.4 t + 1.5$ , for $t \geq 0$ , until it lands.

(a) Write down the height from which the rocket is launched. [1]

(b) Find the maximum height of the rocket. [2]

(d) Find the length of time for which the rocket is more than 30 metres above the ground. [2]

Show the markscheme

(a) $1.5$ m A1

(b) attempt to find the maximum, e.g. sketch of the graph with the maximum marked, or $t = - \frac{29.4}{2 \times (- 4.9)} = 3$ (M1)

$45.6$ m A1

$t = 6.05$ s $(6.05059 \dots)$ A1

Note: Award (M1)A0 for $t = 6.05$ and $t = - 0.0506$ both given as final answers without rejecting the negative value.

(d) $- 4.9 t^{2} + 29.4 t + 1.5 = 30$ or a sketch with $y = 30$ drawn, giving $t = 1.21571 \dots$ and $t = 4.78428 \dots$ (M1)

$4.78428 \dots - 1.21571 \dots = 3.57$ s $(3.56857 \dots)$ A1

Total [7]

Part (a) is a "write down" because the launch height is the value of $h$ at $t = 0$ , which is the constant term. Parts (b) to (d) are all GDC parts, and in each the method mark is for the equation or the graph. In part (d) both intersections must be found from unrounded values: subtracting the rounded values 4.78 and 1.22 gives 3.56, which is wrong to 3 significant figures.

The next question uses a system of equations, one of the commonest set-ups on the paper.

The three marks in part (a) are for the set-up alone, which is typical. If one equation in (a) is wrong, part (b) can still earn follow-through marks for solving the candidate's own system correctly, which is only possible if the system is written down.

Statistics is the largest topic on the paper, and the chi-squared test is one of its most frequent shapes. Its parts are short and mostly "write down", so the marks depend on exact wording. The next question is based on a hypothetical survey of 150 students, whose results are shown by year group.

	Library	Home	Café
Year 12	28	35	12
Year 13	18	40	17

A school asks 150 students where they prefer to study. The results, by year group, are given in the table of study locations.

A chi-squared test for independence is carried out at the 5% significance level.

(a) Write down the null hypothesis. [1]

(b) Write down the number of degrees of freedom. [1]

(d) Write down the p-value for this test. [2]

(e) State the conclusion of the test, giving a reason for your answer. [2]

Show the markscheme

(a) $H_{0}$ : study location preference is independent of year group A1

(b) $ν = (2 - 1) (3 - 1) = 2$ A1

$= 23$ AG

(d) $0.186$ $(0.185507 \dots)$ A2

(e) $0.186 > 0.05$ R1

there is insufficient evidence to reject $H_{0}$ : study location preference appears to be independent of year group A1

Note: Do not award R0A1.

Total [7]

Part (c) shows how a "show that" works on a calculator paper. The answer 23 is given, so writing "23" earns nothing; the mark is for the calculation that produces it, row total times column total divided by the grand total. Part (d) is worth 2 marks for a single number because the whole test is done on the GDC, and the mark is for the right test with the right data. In part (e) the reason (the comparison of the p-value with 5%) earns the R1, and the conclusion without it earns nothing.

Finance questions on Paper 1 are short and almost always use the TVM solver, so the marks depend on the entries and on the rounding the question asks for.

Keiko invests $8000 in an account that pays a nominal annual interest rate of 3.6%, compounded monthly.

(a) Find the value of the investment after 5 years. Give your answer to the nearest dollar. [3]

(b) Find the number of complete years until the value of the investment first exceeds $12,000. [3]

Show the markscheme

(a) EITHER

$N = 60, I % = 3.6, P V = - 8000, P M T = 0, P / Y = 12, C / Y = 12$ (M1)(A1)

$8000 {(1 + \frac{3.6}{100 \times 12})}^{12 \times 5}$ (M1)(A1)

THEN

$$ 9575$ $(9575.15 \dots)$ A1

Note: Award A0 for an answer not given to the nearest dollar, or given as a negative value.

(b) $I % = 3.6, P V = - 8000, P M T = 0, F V = 12 000, P / Y = 12, C / Y = 12$ (M1)

$N = 135.357 \dots$ months, or $n = 11.2798 \dots$ years (A1)

12 years A1

Total [6]

Part (b) needs care with the context. The solver gives about 11.28 years, but the value only exceeds $12,000 during the 12th year: after 11 complete years it is $11,879.91 and after 12 it is $12,314.71. "Complete years" means the answer is rounded up, not to the nearest whole number. The sign convention (money paid in is entered as negative, money received as positive) and the full entries are taught in subtopic 1.4, Financial applications of geometric sequences.

Common mistakes on Paper 1

Many of the marks lost on Paper 1 come from a small number of habits, and the examiners' reports for this course return to the same ones each session.

Answers in calculator notation are a related loss. A value displayed as 1.14E10 must be written as $1.14 \times 10^{10}$ ; calculator notation can imply a method mark but never earns the accuracy mark. The same applies to typing syntax such as binomcdf(20,0.3,4) in place of a probability statement.

Units and context are the last of the common losses. Give units with every answer that has them (m, s, ${cm}^{2}$ , $), and make sure the answer is possible in context: a number of people is a whole number, a probability is between 0 and 1, and a time cannot be negative.

Exam tips for Paper 1

The first read of a question should take about 20 seconds: identify the shape, underline the command term of each part and the accuracy asked for, and note which parts depend on earlier ones. A part that depends on an earlier answer is the place where a "show that" or a given value lets you continue even if the earlier part went wrong.

At the end of the paper, the most valuable checks are fast ones: is every part attempted, is every final answer to 3 significant figures (or the accuracy asked for), are units there, and has every negative or impossible root been rejected? Re-doing long calculations is a poor use of the last few minutes.

Summary

Paper 1 is a set of compulsory short questions on one context each, 13 questions on SL papers, usually 4–9 marks each, covering every topic.
The command term at the start of each part sets the kind of answer and the working needed; "write down" needs none, "find" and "calculate" need stages.
Budget about a mark a minute (1.125 min per mark at SL, about 1.09 at HL); skip a stalled part and return to it.
Most method marks are for the set-up: the equation, system, distribution, TVM entries or test, written down before the GDC answer.
Bracketed marks are implied by a correct answer; unbracketed marks need the working seen; a wrong answer with no working earns nothing.
Keep full values in the GDC, round only final answers, and never leave calculator notation.
Read the last sentence of each part for the accuracy, form and counting instruction.
Statistics and probability is the largest topic on SL Paper 1, with geometry and trigonometry and functions close behind.

Self-check

To do:
I can name the command term of a part and say how much working it needs.
To do:
I can recognise the common Paper 1 shapes from the stem and plan the GDC tool for each part.
To do:
I can work out a time budget for any question from its marks.
To do:
I can write the set-up line the markscheme expects for an equation, a system, a TVM calculation, a probability and a test.
To do:
I can explain the difference between (M1) and M1 in a markscheme.
To do:
I can reject an impossible root and say why in a few words.
To do:
I can round correctly for "complete years", "nearest dollar" and "3 significant figures".
To do:
I can convert a calculator display such as 3.2E−4 into correct notation.

Carry on with Maths AI in Augment

With a free account, Augment remembers where you stopped in the notes and which questions you have marked.

Start free