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IB Maths AI HL Drilling Method & Past Paper Practice (Paper 1): 10 GDC Short-Response Questions
TUTORZONE ORIGINAL MOCK PAPER · IB MATHEMATICS AI (APPLICATIONS AND INTERPRETATION) HL PAPER 1
The biggest difference between IB Maths AI (Applications and Interpretation) and AA is that every AI paper allows the GDC: drilling is about turning a real context into a mathematical model and interpreting the result in that context. AI HL: Paper 1 (technology) 110 marks in 2 h at 30%, Paper 2 110 marks in 2 h at 30%, Paper 3 (HL only) 55 marks in 1 h at 20%, and the exploration 20%. This set has 10 short-response practice questions in the Paper 1 style (80 marks in this set) covering depreciation, statistics, probability, the normal distribution, geometry, calculus optimisation, series and regression, each with its own model, computation and interpretation.
What is in IB Maths AI HL Paper 1?
Direct answer: IB Maths AI (Applications and Interpretation) HL Paper 1 allows the GDC and carries 110 marks over 2 hours at 30% of the subject, entirely compulsory short-response; Paper 2 also allows the GDC at 110 marks, 2 hours, 30%; Paper 3 (HL only) is 55 marks, 1 hour, 20%; and the exploration is 20%. The key contrast with AA is that every AI paper allows the GDC, so drilling is about modelling and interpretation rather than exact hand algebra.
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Paper 1 (technology allowed) | Compulsory short-response on the whole syllabus; HL 110 marks, 2 h, 30% | HL 30% / SL 40% | 2 h |
| Paper 2 (technology allowed) | Compulsory extended-response; HL 110 marks, 2 h, 30% | HL 30% / SL 40% | 2 h |
| Paper 3 (HL only) | Two compulsory extended problem-solving questions; 55 marks, 1 h, 20% | HL 20% (HL only) | 1 h |
| Mathematical exploration (IA) | 20% at both levels; 15 recommended hours | 20% (SL and HL) | 15 recommended hours |
The official HL Paper 1 runs 120 minutes for 110 marks; this set’s 10 questions work out at about 12 minutes each. Marks in AI are rarely lost to arithmetic but to pressing the GDC before writing the model: a bare number with no equation, unit or contextual comment loses the method marks.
How this paper was built
This is a TutorZone original mock paper: the questions follow the AI structure (HL Paper 1, technology allowed, 30% over 2 hours) and are our own writing, not IBO past papers or official marking references.
Ten short-response questions for 80 marks in this set (the official HL Paper 1 runs 110 marks over 2 hours); every numerical answer was recomputed programmatically and is given to 3 significant figures, as AI papers expect.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Paper 1 (technology allowed) | HL 30% / SL 40% | about 110 min with 10 min to check | write the model before reaching for the GDC |
| Paper 2 (technology allowed) | HL 30% / SL 40% | about 20 min per extended question | show the derivation and the interpretation |
| Paper 3 (HL only) | HL 20% (HL only) | about 26 min per question | write the assumptions and route first |
| Mathematical exploration (IA) | 20% (SL and HL) | topic, model, computation, interpretation | pick a question you can model with data |
How many questions does this paper have, and how should the time be split?
Direct answer: 10 short-response questions for 80 marks in this set (the official HL Paper 1 is 110 marks in 2 hours); aim for 110 minutes with 10 to check (about 12 minutes each). Give every answer to 3 significant figures unless the question says otherwise.
Paper 1: short-response questions with the GDC (10 questions, 80 marks)
All compulsory; aim for 110 minutes with 10 to check. Give answers to 3 significant figures unless told otherwise.
Question 1 (8 marks)
A machine is bought for $250 000 and depreciates by 15% each year on a reducing-balance basis.
(a) Find its value after 4 years (3 s.f.). (b) Find the first year in which its value falls below $100 000.
Question 2 (8 marks)
The delivery times (minutes) of six orders were 14, 18, 21, 25, 30 and 36.
(a) Find the mean and the sample standard deviation (3 s.f.). (b) Find the median and say what its comparison with the mean suggests.
Question 3 (8 marks)
In a survey of 200 students, 120 use a tablet, 90 use a laptop and 60 use both.
(a) Find P(uses a tablet | uses a laptop). (b) Find the probability that a randomly chosen student uses exactly one of the two devices.
Question 4 (8 marks)
A filling machine delivers a volume that is normally distributed with mean 500 ml and standard deviation 8 ml.
(a) Find P(X < 492). (b) Find the volume below which 2.5% of bottles fall (3 s.f.).
Question 5 (8 marks)
A cylindrical water tank has radius 1.2 m and height 2.5 m.
(a) Find its volume in cubic metres (3 s.f.) and in litres. (b) Find the area of the curved surface (3 s.f.).
Question 6 (8 marks)
A closed box has a square base of side x cm and a volume of 500 cm³.
(a) Show that its total surface area is S = 2x² + 2000 ÷ x. (b) Find the x that minimises S and the minimum S (both 3 s.f.).
Question 7 (8 marks)
An online shop’s sales were $120 000 in its first year and rise by $8 000 each year.
(a) Find its sales in year 10. (b) Find total sales over the first 10 years.
Question 8 (8 marks)
A drone’s height above the ground t seconds after take-off is h(t) = −5t² + 20t + 10 metres (0 ≤ t ≤ 5).
(a) Find the maximum height and when it occurs. (b) Find the two times at which the drone is 10 m above the ground and explain what they mean.
Question 9 (8 marks)
A culture starts with 500 bacteria and grows by 35% each hour.
(a) Find the number after 6 hours (nearest integer). (b) Find the time to reach 5 000 bacteria and the doubling time (hours, 3 s.f.).
Question 10 (8 marks)
Five students’ weekly study hours x and test scores y are (1, 42), (2, 50), (3, 61), (4, 66) and (5, 76).
(a) Find Pearson’s correlation coefficient r (3 s.f.). (b) Find the least-squares regression line y = a + bx (a and b to 3 s.f.) and predict the score for 6 hours of study.
Where are the answers, and do they show the working?
Direct answer: every question is followed by an answer block giving the model, the step-by-step computation and the interpretation, with the marks for each step. The depreciation item, for example, splits into 2 for the model, 3 for the computation and 3 for deciding the year – not a bare number.
Paper 1 answers (model, computation, interpretation)
Question 1
(1) Model: V = 250 000 × 0.85ⁿ (2 marks); (2) after four years 250 000 × 0.85⁴ = $131 000 (3 s.f.) (3 marks); (3) solving 0.85ⁿ < 0.4 gives n > 5.64, so the value first falls below $100 000 in year 6 (3 marks). Year 5 is still about $111 000, so the years either side must be evaluated to decide.
Question 2
(1) Mean = 144 ÷ 6 = 24 minutes (3 marks); (2) sample standard deviation = √(326 ÷ 5) = 8.07 (3 s.f.) (3 marks); (3) median = (21 + 25) ÷ 2 = 23 minutes; the mean exceeds the median, so the data are slightly positively skewed (2 marks). The sample standard deviation divides by n − 1 = 5; only a population question divides by n.
Question 3
(1) Conditional probability = 60 ÷ 90 = 2/3 (4 marks); (2) exactly one = (120 − 60) + (90 − 60) = 90 students, so 90 ÷ 200 = 0.45 (4 marks). The denominator of a conditional probability is the condition group (90), not the whole sample (200).
Question 4
(1) z = (492 − 500) ÷ 8 = −1, so P = 0.159 (3 s.f.) (4 marks); (2) 2.5% corresponds to z = −1.96, so x = 500 − 1.96 × 8 = 484.32 ≈ 484 ml (4 marks). State the z-value with its sign; even with the GDC’s normal function the population σ (8) is used, not a sample deviation.
Question 5
(1) V = π × 1.2² × 2.5 = 11.3 m³ (3 s.f.), i.e. 11 300 litres (4 marks); (2) curved surface area = 2π × 1.2 × 2.5 = 18.8 m² (4 marks). 1 m³ = 1000 litres; the curved area is 2πrh and excludes the two ends.
Question 6
(1) With height h, x²h = 500 so h = 500 ÷ x²; S = 2x² + 4xh = 2x² + 2000 ÷ x (3 marks); (2) dS/dx = 4x − 2000 ÷ x² = 0 gives x³ = 500, x = 7.94 cm (3 marks); (3) S = 6x² = 378 cm² (2 marks). dS/dx = 0 only gives a stationary point – add that the second derivative is positive, so it is a minimum. At the optimum h = x, so the box is a cube.
Question 7
(1) u₁₀ = 120 000 + 9 × 8 000 = $192 000 (4 marks); (2) S₁₀ = 10 ÷ 2 × (120 000 + 192 000) = $1 560 000 (4 marks). Year 1 to year 10 is nine increases, not ten; the average method for an arithmetic sum is (first + last) ÷ 2 × number of terms.
Question 8
(1) The vertex is at t = −20 ÷ (2 × −5) = 2 s, h(2) = −20 + 40 + 10 = 30 m (4 marks); (2) solving −5t² + 20t + 10 = 10, i.e. −5t (t − 4) = 0, gives t = 0 s and 4 s (2 marks); (3) t = 0 is the take-off height and t = 4 s is the moment it returns to that height (2 marks). Setting h = 10 cuts the parabola twice; only the second root is the return.
Question 9
(1) Model N = 500 × 1.35ᵗ, so N(6) = 3 027 (3 marks); (2) 500 × 1.35ᵗ = 5 000 means 1.35ᵗ = 10, so t = ln10 ÷ ln1.35 = 7.67 hours (3 marks); (3) doubling time = ln2 ÷ ln1.35 = 2.31 hours (2 marks). Take logarithms of both sides explicitly; the doubling time depends on the growth rate, not on the starting count.
Question 10
(1) x̄ = 3, ȳ = 59, Sxy = 84, Sxx = 10, Syy = 712, so r = 84 ÷ √(10 × 712) = 0.995 (3 marks); (2) b = 84 ÷ 10 = 8.4 and a = 59 − 8.4 × 3 = 33.8, giving y = 33.8 + 8.4x (3 marks); (3) at x = 6, y = 33.8 + 50.4 = 84.2 marks (2 marks). An r near 1 means a strong linear association, not licence to extrapolate – 6 hours is already just outside the data range of 1 to 5 hours.
Where are marks most easily lost in an AI paper?
Direct answer: copying the GDC display without writing a model; the wrong number of significant figures (130501.5625 where 3 s.f. is asked); and extrapolating a regression without checking the data range. All three cost method and interpretation marks.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Copying the GDC display | Only 131000 is written | No model means no method marks | Write 250 000 × 0.85⁴ first, then the value |
| Wrong number of significant figures | Writing 130501.5625 | AI expects 3 s.f., so over-long answers lose marks | Write 131 000 and state 3 s.f. |
| Ignoring the data range in regression | Treating the x = 6 prediction as exact | A high r is not a licence to extrapolate | State the 1 to 5 hour range, then comment on the extrapolation |
How to use this paper
First sit: no references, the full 120 minutes, using the GDC – but write the model for every question before computing.
Mark it: check the model, the units and the significant figures item by item, especially the contextual sentence.
Second sit: redo the paper three to seven days later until model, computation and interpretation are all present and the time is shorter.
What should I do after finishing this paper?
Direct answer: log every question where the model was missing; redo the paper three to seven days later until model, computation and interpretation are all there; then drill Paper 2 extended-response and the two Paper 3 problem-solving questions by topic.
FAQ
Is this an IBO paper?
No. The questions, answers and mark allocation are TutorZone originals, not IBO past papers or marking references.
How does AI differ from AA?
Every AI paper allows the GDC, so drilling centres on modelling and interpretation; AA Paper 1 bans the calculator and wants exact values (surd, fraction, pi).
How many papers does AI HL have?
Three: Paper 1 at 30% (2 hours), Paper 2 at 30% (2 hours) and Paper 3 at 20% (1 hour), plus the exploration at 20%.
Official references
| Resource | What to check |
|---|---|
| IBO Mathematics AI subject brief (first assessment 2021) | Confirm the SL/HL weightings, durations and the technology policy |
| https://www.tutorzone.com.hk/en-ib-math-ai-past-paper-drill-guide/ | IB Maths AI drilling guide: modelling and interpretation |
| https://www.tutorzone.com.hk/en-ib-math-ai-sl-mock-paper/ | IB Maths AI SL mock (same series) |
Related: IB practice area · IB Maths AI drilling guide · IB Maths AI SL mock · IB Maths AI SL Paper 2 (GDC)
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