60 Maths IA Research Questions

Ready-to-use IB Mathematics IA ideas for AI SL, AI HL, AA SL and AA HL — each with a free data source and a worked mathematical approach.

Stay within your course.

Every question here is scoped to one course level. You won't be penalised for using maths beyond your syllabus, but it's a common way to end up with a harder, weaker IA. The maths you've been taught is more than enough for full marks.

I wrote these by working through every Paper 3 from the new syllabus (2021–2024). Each idea takes a theme from one of those papers — cubic polynomials, AM-GM inequalities, logistic virus models, Voronoi diagrams — and turns it into a research question. None of them need a student survey: the data is either free to download or easy to collect at home.

How to use this page

  1. Find something that appeals. Pick your course below and browse. You're looking for a topic that sparks genuine curiosity — don't overthink it yet.
  2. Check the data and the maths. Can you realistically get the data? Most sources download in under ten minutes. If you can follow the mathematical approach, the topic is probably right for you.
  3. Try the Paper 3 that inspired it. If you can answer those questions confidently, the maths is within your grasp — and it counts as good research for your IA.
  4. Make it personal, but stay within your course. Personal engagement is one of the hardest marks to earn. Go beyond the exam paper by changing the context to something that matters to you, not by reaching for harder maths.

Applications & Interpretation — Standard Level 20 questions

AI SL#1

How well does a sinusoidal model T(m) = a·sin(bm + c) + d describe the monthly variation in daylight hours in a chosen city, and what do the parameters reveal about the city’s latitude?

Data
timeanddate.com — enter any city to get monthly hours of daylight; free, no login required.
Mathematical approach
Collect monthly daylight data; use GDC sinusoidal regression to fit the model; interpret amplitude a as half the annual range and d as the average daily hours; assess R²; compare two cities at different latitudes and interpret the parameter differences.
Inspired by
IB AI HL Paper 3, May 2021
AI SL#2

Is there a statistically significant linear relationship between atmospheric CO₂ concentration and global mean temperature anomaly, and does the strength of this relationship differ between the pre-1980 and post-1980 periods?

Data
NASA GISS Surface Temperature Analysis (data.giss.nasa.gov) and NOAA Mauna Loa CO₂ data (gml.noaa.gov/ccgg/trends) — both freely downloadable as annual CSV files.
Mathematical approach
Pair annual CO₂ (ppm) with temperature anomaly (°C) for each year; compute Pearson’s r and test H₀: ρ = 0 using a t-statistic; fit regression T = a + b·CO₂; split the dataset at 1980 and compare r-values for each half; plot residuals and interpret.
Inspired by
IB AI HL Paper 3, May 2021
AI SL#3

How accurately does an exponential model describe the early growth of a social media post’s views, and at what point does a logistic model provide a better fit?

Data
Student’s own YouTube or TikTok analytics dashboard (accessible if you have a channel), or the publicly available Kaggle “YouTube Trending Videos” dataset (kaggle.com) for real view-count data.
Mathematical approach
Collect cumulative view data over time; linearise with ln(V) vs t to fit V = Ae^(kt); also fit logistic V = L/(1 + Ce^(−kt)) using GDC; compare R² values and residual plots; identify the inflexion point and interpret the carrying capacity L.
Inspired by
IB AI HL Paper 3, May 2022
AI SL#4

Do the leading digits of reported COVID-19 cumulative case counts across countries follow Benford’s Law, and what does a chi-squared test reveal about potential reporting anomalies?

Data
Our World in Data COVID-19 dataset (ourworldindata.org/coronavirus) — free CSV download with cumulative confirmed cases by country; no registration required. Use the most recent value per country.
Mathematical approach
Extract the leading digit of each country’s cumulative case count (100+ countries); compute observed frequency distribution; compare against Benford’s expected proportion P(d) = log₁₀(1 + 1/d) for d = 1,...,9; conduct χ² goodness-of-fit test with 8 degrees of freedom; compute standardised residuals to identify which digits deviate most.
Inspired by
IB AI HL Paper 3, May 2021
AI SL#5

How does the optimal placement of a new café — determined using Voronoi diagrams — change when foot-traffic weighting is introduced, and how much does the expected customer travel distance decrease?

Data
Google Maps — plot existing cafés and record their coordinates using the right-click tool; foot-traffic estimates from Google’s ‘Popular Times’ feature or publicly available pedestrian count surveys.
Mathematical approach
Plot existing cafés as Voronoi sites on a coordinate grid; identify the largest Voronoi cell; find the circumcentre of its bounding triangle as the unweighted optimal site; apply demand-weighted Voronoi adjustment; compute average expected travel distances before and after and quantify the improvement.
Inspired by
IB AI HL Paper 3, May 2022
AI SL#6

How accurately does a sinusoidal model describe the cyclical fluctuations in snowshoe hare and lynx populations in Canada, and what does the phase difference between the two cycles reveal about predator-prey dynamics?

Data
Hudson Bay Company historical lynx and hare fur pelt records (1845–1935) — freely available from academic sources including people.whitman.edu/~balajid/data/LynxHare.csv. No registration required.
Mathematical approach
Fit H(t) = a₁sin(b₁t + c₁) + d₁ and L(t) = a₂sin(b₂t + c₂) + d₂ using GDC sinusoidal regression; compare R²; compute the phase lag c₂ − c₁ (how many years the lynx peak lags behind the hare peak); verify that period 2π/b is consistent with the known 9–11 year ecological cycle; interpret biologically.
Inspired by
IB AI HL Paper 3, May 2021
AI SL#7

Which regression model — linear or quadratic — best predicts a car’s stopping distance at different speeds, and which model is more physically meaningful?

Data
UK Highway Code Table of overall stopping distances (gov.uk/guidance/the-highway-code/general-rules-techniques-and-advice-for-all-drivers-and-riders-103-to-158) — seven speed/distance pairs, officially published and freely available.
Mathematical approach
Fit d = a + bv (linear) and d = a + bv + cv² (quadratic) using a GDC; compare R²; argue from kinetic energy (KE ∝ v²) that the quadratic is physically correct; predict stopping distance at 100 km/h and compare with observed values; interpret residuals.
Inspired by
IB AI HL Paper 3, May 2024
AI SL#8

How does the number of ‘hops’ needed for information to spread through a social network depend on the network’s structure, and what does the adjacency matrix reveal?

Data
Student-constructed from own friendship or class WhatsApp network (15–20 people). Map mutual connections on a labelled graph and transcribe into an adjacency matrix. No external data needed.
Mathematical approach
Construct adjacency matrix A for a chosen network; compute A², A³ until all off-diagonal entries are non-zero; compare the minimum hop-count for tree, cycle, and complete graph structures of the same size; discuss how network density affects spreading speed.
Inspired by
IB AI HL Paper 3, May 2022
AI SL#9

Can the spread of seasonal influenza through a school population be modelled by discrete SIR difference equations, and how sensitive is the epidemic peak to the infection rate β and recovery rate γ?

Data
UK flu surveillance reports (gov.uk/government/collections/weekly-national-flu-reports) — historical weekly GP consultation rates by age group, freely downloadable as Excel/CSV.
Mathematical approach
Set up S_{t+1} = S_t − βS_tI_t; I_{t+1} = I_t + βS_tI_t − γI_t; R_{t+1} = R_t + γI_t; calibrate β and γ using real data; iterate using a spreadsheet; plot I(t); find the peak; compute R₀ = β/γ; explore how changing β ± 0.1 shifts the peak timing and magnitude.
Inspired by
IB AI HL Paper 3, May 2022
AI SL#10

Does the number of wins achieved by a football team in non-overlapping blocks of four consecutive matches follow a binomial distribution B(4, p), and does a chi-squared goodness-of-fit test support this model?

Data
football-data.co.uk — free CSV download of match results for the Premier League, La Liga, Serie A, and others, dating back to the 1990s. No registration required. Download a season with at least 200 matches.
Mathematical approach
Group match results into non-overlapping blocks of 4; count wins per block; estimate p = (total wins)/(total matches); compute expected binomial frequencies; merge any cell below 5; conduct χ² test with (categories − 2) degrees of freedom; interpret and discuss whether the independence assumption holds across the season.
Inspired by
IB AI HL Paper 3, Nov 2022
AI SL#11

If a patient takes a 200 mg dose of ibuprofen every 8 hours, and the drug has a published elimination half-life of approximately 2 hours, how does the peak blood concentration evolve across successive doses and what is the theoretical steady-state peak?

Data
No external dataset required. The elimination half-life of ibuprofen (t½ ≈ 2 hours) is a published pharmacological constant, documented in the NHS British National Formulary (bnf.nice.org.uk).
Mathematical approach
Set k = ln2/t½; model C_n⁺ = d + C_{n-1}⁺ · e^{−kT} where T = 8 h; iterate for n = 1,...,10 and plot; identify the sequence as geometric; find C_∞ = d/(1 − e^{−kT}); determine what percentage of steady state is reached after 5 doses; find the minimum T keeping trough concentration above a stated therapeutic threshold.
Inspired by
IB AI HL Paper 3, Nov 2022
AI SL#12

Is there a statistically significant difference in mean annual global temperature anomaly between the 1970s and the 2010s, and does a two-sample t-test support the claim that recent decades are significantly warmer?

Data
NASA GISS Surface Temperature Analysis (data.giss.nasa.gov/gistemp/tabledata_v4/GLB.Ts+dSST.csv) — free annual global temperature anomaly data. Use the J-D (annual mean) column. No login required.
Mathematical approach
Extract the 10 annual anomaly values for 1970–1979 and 2010–2019; compute sample means and standard deviations; state H₀: μ₁ = μ₂; perform a two-sample t-test using a GDC; calculate effect size (Cohen’s d); compare statistical significance with practical significance.
Inspired by
IB AI HL Paper 3, May 2021
AI SL#13

How does the normal distribution model the global distribution of total IB Diploma scores, and what fraction of candidates fall more than two standard deviations from the mean?

Data
IB Statistical Bulletin (ibo.org/about-the-ib/facts-and-figures/statistical-bulletins) — published annually as a free PDF. Reports global mean, standard deviation, and grade frequency tables for each May session.
Mathematical approach
Source global mean μ and standard deviation σ from the Statistical Bulletin; standardise to z-scores; overlay a normal curve on the published grade frequency data; compute P(|X − μ| > 2σ) theoretically and compare with the empirical proportion; discuss whether the normal assumption is justified for integer-valued scores.
Inspired by
IB AI HL Paper 3, May 2021
AI SL#14

How accurately does Newton’s Law of Cooling model the temperature drop of a cup of coffee, and how does the decay constant k differ between a ceramic mug and a paper cup?

Data
Student-collected: a thermometer (or digital temperature probe) and stopwatch. Record temperature every 60 seconds for 20–30 minutes for each vessel. Reproducible in any kitchen — no specialist equipment needed.
Mathematical approach
For each cup, fit T(t) = T_room + (T₀ − T_room)e^(−kt) by linearising to ln(T − T_room) vs t; find k by linear regression on the GDC; compare k-values and interpret in terms of insulation; assess whether deviations from linearity at early or late times suggest a more complex model is needed.
Inspired by
IB AI HL Paper 3, May 2023
AI SL#15

How well does a logistic model fit the cumulative COVID-19 vaccination rollout in a chosen country, and what does the inflexion point reveal about when the campaign was at its fastest?

Data
Our World in Data (ourworldindata.org/covid-vaccinations) — free CSV download of cumulative vaccinations per hundred people by country and date. No registration required. Choose a country that completed its primary rollout.
Mathematical approach
Fit logistic V(t) = L/(1 + Ce^{−kt}) to cumulative percentage vaccinated vs time using GDC; interpret L as the maximum rollout coverage; find the inflexion point at t* = ln(C)/k (point of fastest uptake); compare R² with a pure exponential fit; discuss the policy implications of the parameters.
Inspired by
IB AI HL Paper 3, May 2022
AI SL#16

How does the probability of winning a best-of-five sports series change as the per-game win probability p varies, and how much does a small home advantage amplify a team’s overall series win probability?

Data
Pure mathematical model — no data collection required. Results can optionally be compared with historical NBA Finals or World Series data (basketball-reference.com or baseball-reference.com).
Mathematical approach
Compute P(win series) = P(X ≥ 3) where X ∼ B(5, p); plot P_series vs p on a GDC; compute the ‘amplification ratio’ P_series/p; find the p that maximises this ratio; extend the analysis to best-of-seven and compare how the amplification changes.
Inspired by
IB AA HL Paper 3, May 2024
AI SL#17

Is the relationship between a country’s GDP per capita and its average life expectancy better described by a linear or a logarithmic regression model, and what does this imply about the diminishing returns of economic growth on human longevity?

Data
World Bank Open Data (data.worldbank.org/indicator) — download life expectancy (SP.DYN.LE00.IN) and GDP per capita (NY.GDP.PCAP.CD) for 150+ countries in any single year. Free CSV, no registration required.
Mathematical approach
Plot life expectancy L against GDP per capita G; note the curved relationship; fit L = a + bG (linear) and L = a + b·ln(G) (logarithmic) using GDC; compare R² and residual plots; test significance of the slope with a t-test; interpret: what does a doubling of GDP per capita predict for life expectancy?
Inspired by
IB AI HL Paper 3, May 2024
AI SL#18

For a basketball free throw, what combination of initial speed and launch angle gives the largest ‘scoring window’ — the range of angles that successfully sends the ball through the hoop?

Data
Pure mathematical model using official FIBA/NBA specifications: ball diameter 23.8 cm, hoop diameter 45 cm, hoop height 3.05 m, free throw line distance 4.57 m from the hoop centre.
Mathematical approach
Model trajectory y = x·tanθ − gx²/(2v²cos²θ); require y(x_hoop) to lie within [h_rim, h_rim + (d_hoop − d_ball)]; solve for the range of valid θ at each initial speed v; find the v that maximises the angular window; compare with the angle elite players typically use (≈50–55°).
Inspired by
IB AI HL Paper 3, May 2024
AI SL#19

Are road traffic accidents in Great Britain uniformly distributed across days of the week, and what does a chi-squared goodness-of-fit test reveal about which days are statistically more or less dangerous?

Data
UK Department for Transport STAT19 road accident data (data.gov.uk/dataset/cb7ae6f0-4be6-4935-9277-47e5ce24a11f/road-safety-data) — free annual CSV, updated each year. US alternative: NHTSA FARS data (nhtsa.gov/research-data/fatality-analysis-reporting-system-fars).
Mathematical approach
Count total reported accidents by day of week for a chosen year; compute expected frequency = total/7 under H₀ of uniform distribution; conduct χ² goodness-of-fit test with 6 degrees of freedom; compute standardised residuals (O−E)/√E; extend by comparing weekday vs weekend accident severity profiles.
Inspired by
IB AI HL Paper 3, Nov 2022
AI SL#20

How does exponential depreciation compare to straight-line depreciation for a car’s resale value over 10 years, and which model better fits real asking-price data for a popular model?

Data
AutoTrader.co.uk or CarGurus.com — search for a specific make and model (e.g. Ford Focus) and record asking price vs year of registration for 20–30 listings. Takes approximately 20 minutes.
Mathematical approach
Record (age, price) pairs; fit V(t) = V₀e^(−kt) by linearising ln(V) vs t; also fit V(t) = V₀ − mt by linear regression; compute R² for both; plot residuals; discuss when each model predicts zero value; interpret the exponential’s ‘never reaches zero’ property and its implications for insurance.
Inspired by
IB AI HL Paper 3, May 2023

Applications & Interpretation — Higher Level 10 questions

AI HL#1

How accurately can R(t) = A·sin(ωt) + B·cos(ωt) model the time of sunrise in a given city, and how does converting this to a single sinusoid reveal the amplitude and phase of the seasonal cycle?

Data
timeanddate.com — provides exact sunrise times for any city on any date, free, no login required. Download monthly sunrise times for at least two full years.
Mathematical approach
Fit R(t) = A·sin(ωt) + B·cos(ωt) to sunrise-time data; convert to R·sin(ωt + φ) using R = √(A² + B²) and tanφ = B/A; interpret R as the seasonal amplitude and φ as the date of latest sunrise; compare R² for the combined form vs a single sinusoid at two different latitudes.
Inspired by
IB AI HL Paper 3, May 2021
AI HL#2

Which model — exponential, logistic, or Gompertz — best fits the cumulative confirmed case data for a historical disease outbreak, and what do the parameters represent epidemiologically?

Data
Our World in Data (ourworldindata.org/coronavirus) — cumulative confirmed case counts by country and date, free CSV download. Use a single-wave outbreak for cleaner fitting (e.g. a specific country during the initial COVID-19 wave).
Mathematical approach
Fit Q(t) = Ae^(bt), Q(t) = L/(1 + Ce^{−kt}), and Q(t) = Le^{−ae^{−bt}} to the data; compare R² or residual sum of squares; identify inflexion points analytically (logistic: t* = ln(C)/k; Gompertz: t* = ln(a)/b); interpret L as the saturation level and discuss which model is most appropriate.
Inspired by
IB AI HL Paper 3, May 2022
AI HL#3

For a set of cities, how does the optimal location for a new distribution hub — found using Voronoi diagrams — change when cities are weighted by their annual delivery demand?

Data
Google Maps for city coordinates (right-click to get latitude/longitude); demand weights from publicly available city population data (ons.gov.uk for UK, or World Bank for international). Hypothetical weights also acceptable.
Mathematical approach
Set up unweighted Voronoi cells; find the circumcentre of the largest-gap triangle as the unweighted optimal site; introduce demand weights and adjust using the weighted Voronoi construction; compute total expected delivery distance before and after, and quantify the percentage improvement.
Inspired by
IB AI HL Paper 3, May 2022
AI HL#4

For a barrel modelled as the solid of revolution of a parabola rotated about the x-axis, what shape of parabola maximises volume for a fixed diagonal length (inspired by Kepler’s wine-barrel method)?

Data
Pure mathematical model — no external data required. Historical context from Kepler’s Nova Stereometria Doliorum (1615).
Mathematical approach
Express V = π∫₀^L [f(x)]² dx for f(x) = ax² + bx + c satisfying end-circle and diagonal constraints; write a as the free parameter; differentiate V with respect to a and set to zero; find the optimal a; compare volumes of the optimal parabolic barrel vs the best-fitting cylinder.
Inspired by
IB AI HL Paper 3, May 2023
AI HL#5

How does the minimum separation distance between two aircraft on intersecting flight paths evolve over time, and can a conflict be avoided by adjusting one aircraft’s speed by the smallest possible amount?

Data
Pure mathematical model using realistic position and velocity vectors. Optional: real track data from the OpenSky Network (opensky-network.org), which provides free historical ADS-B flight data.
Mathematical approach
Write position vectors r_A(t) = p_A + t·v_A and r_C(t) = p_C + t·v_C; form d²(t) = |r_A − r_C|²; differentiate and set to zero to find t_min; compute minimum distance; if below 10 km, find the minimum speed adjustment to v_C that raises the separation above 10 km.
Inspired by
IB AI HL Paper 3, May 2023
AI HL#6

How does the power of a one-tailed binomial hypothesis test change with sample size, and what is the minimum n needed to achieve 80% power against a specified alternative hypothesis?

Data
Pure mathematical exploration — no external data required. Power curve calculations performed using the GDC binomial CDF function.
Mathematical approach
Define H₀: p = p₀ vs H₁: p < p₀; find the critical region at α = 0.05; compute power = P(X ≤ c | p = p₁) for n = 10, 20, ..., 100; plot power curves; find the minimum n giving power ≥ 0.80; discuss the Type I vs Type II error trade-off and its practical implications.
Inspired by
IB AI HL Paper 3, May 2024
AI HL#7

For a T-shirt cannon fired at fixed initial speed, what is the area of the ‘reachable region’ bounded by the envelope parabola, and what fraction of a stadium’s seats lie within this region?

Data
Pure mathematical model. Stadium seating dimensions are publicly available from official venue capacity diagrams. T-shirt cannons typically fire at 30–40 m/s.
Mathematical approach
Derive the envelope parabola y = v²/(2g) − gx²/(2v²); compute the area A = ∫ y dx over the ground-level x-range; model seat positions as a sequence; check each seat’s (x, y) against the envelope inequality; compute the reachable fraction.
Inspired by
IB AI HL Paper 3, May 2024
AI HL#8

How does the Spearman’s rank correlation between a footballer’s transfer market value and their goal contributions change between two consecutive seasons, and does this reveal any systematically over- or undervalued players?

Data
Transfermarkt.com (player market valuations by season) and FBref.com (goals + assists per season for top leagues) — both free and publicly searchable. Compile data for the top 50 players in a chosen league.
Mathematical approach
Rank the 50 players by market value and by goal contributions (goals + assists) for each season; compute r_s = 1 − 6Σd²/[n(n²−1)]; test significance using t = r_s√((n−2)/(1−r_s²)); compare r_s across seasons; identify the players whose value and performance rankings diverge most and discuss whether they appear over- or undervalued.
Inspired by
IB AI HL Paper 3, May 2021
AI HL#9

How does the logistic model of a computer-virus infection compare to a simple exponential model, and what does fitting Q’/Q versus Q reveal about the infection rate β and carrying capacity L?

Data
AV-TEST Institute (av-test.org/en/statistics/malware) publishes annual malware registration statistics; alternatively, use time-series infection data from published academic papers on historical malware outbreaks (e.g. Conficker, WannaCry).
Mathematical approach
Start with exponential Q = Ae^(βt); derive the logistic model from Q’ = βQ(1 − Q/L); rearrange to Q’/Q = β − (β/L)Q, which is linear in Q; perform linear regression on empirical Q’/Q vs Q data to estimate β and L; find the time of peak infection rate at Q = L/2.
Inspired by
IB AI HL Paper 3, May 2022
AI HL#10

How does halving the sample size of a contingency table affect the ability of a chi-squared test to detect a genuine association between manufacturing technique and component defect rate, and at what sample size does the test become non-significant?

Data
Pure mathematical exploration starting from a given contingency table with a known significant result. A real-world manufacturing dataset can optionally be sourced from Kaggle (search ‘manufacturing quality’).
Mathematical approach
Start with a contingency table with a known significant result; compute χ² and p-value; scale all frequencies down by factors of 2, 4, 8; re-run the test each time; plot p-value vs sample size; find the threshold n below which the test is non-significant at 5%; compute the Type II error probability at that n.
Inspired by
IB AI HL Paper 3, May 2024

Analysis & Approaches — Standard Level 20 questions

AA SL#1

How does the number of real roots of a cubic polynomial f(x) = x³ − 3cx + d change as the parameter c varies, and can this relationship be characterised using differentiation?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Differentiate to find local max and min; test whether f(local max) > 0 and f(local min) < 0 to determine when three distinct real roots exist; plot the discriminant boundary curve 4c³ = d² in the (c, d) plane; verify with GDC by plotting several curves.
Inspired by
IB AA HL Paper 3, May 2021
AA SL#2

For a regular n-sided polygon, how does the ratio of its area to its perimeter change as n increases, and what value does this ratio approach as n tends to infinity?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Express area A = (ns²/4)cot(π/n) and perimeter P = ns; form the ratio A/P = (s/4)cot(π/n); use the small-angle approximation cot(π/n) ≈ n/π as n → ∞ to show the limiting shape is a circle; verify numerically with a GDC table.
Inspired by
IB AA HL Paper 3, May 2021
AA SL#3

For right-angled triangles with integer side lengths (Pythagorean triples), which triangles have a numerical area equal to their numerical perimeter?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Parametrise Pythagorean triples as (m² − n², 2mn, m² + n²); substitute into A = P; solve the resulting Diophantine equation for integer solutions m and n; verify each case; check whether infinitely many such triangles exist.
Inspired by
IB AA HL Paper 3, May 2021
AA SL#4

How does the area bounded by the curve f(x) = xⁿe⁻ˣ and the x-axis (for x ≥ 0) change as n increases, and is there a pattern that can be conjectured from small cases?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Compute A_n = ∫₀^∞ xⁿe⁻ˣ dx using repeated integration by parts; evaluate for n = 0, 1, 2, 3, 4; conjecture A_n = n! and verify graphically on a GDC.
Inspired by
IB AA HL Paper 3, May 2023
AA SL#5

What is the maximum product of n positive real numbers whose sum is fixed at S, and how does this maximum change as n increases?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Apply the AM-GM inequality to show the maximum is (S/n)ⁿ, achieved when all values equal S/n; verify with specific values; plot M(n) = (S/n)ⁿ for S = 1 and analyse its behaviour as n → ∞.
Inspired by
IB AA HL Paper 3, May 2023
AA SL#6

How does the number of solutions to the equation sin(nx) = x/k change as the integer n and positive constant k are varied?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Plot y = sin(nx) and y = x/k simultaneously on a GDC; count intersections for various (n, k) pairs; derive an approximate counting rule based on how many half-cycles of sin(nx) fit inside [−k, k]; verify the rule for n = 5, 10 and several values of k.
Inspired by
IB AA HL Paper 3, May 2022
AA SL#7

For a compound-interest savings account, how does the effective annual yield differ between monthly, quarterly, and annual compounding, and what is the mathematical limit as compounding frequency tends to infinity?

Data
Pure mathematical exploration. Real interest rates can be sourced from any high-street bank website for a grounding example, but the core investigation requires no data.
Mathematical approach
Use EAR = (1 + r/k)^k − 1 for k = 1, 4, 12, 365; show that as k → ∞ the EAR approaches e^r − 1 (continuous compounding); compare numerically and graphically; discuss when continuous compounding becomes practically indistinguishable from daily compounding.
Inspired by
IB AA HL Paper 3, May 2024
AA SL#8

How many terms of the geometric series with common ratio r are needed to reach at least 99% of the sum to infinity, and how does this number of terms change as r increases from 0 to 1?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Set S_n/S_∞ = 1 − rⁿ ≥ 0.99 and solve for n = log(0.01)/log(r); plot n as a function of r on a GDC; analyse the limiting behaviour as r → 0⁺ and r → 1⁻; extend to the 99.9% case.
Inspired by
IB AA HL Paper 3, May 2024
AA SL#9

How does the optimal launch angle for a projectile change when it is fired from an elevated platform, and how does the optimal angle depend on the platform height?

Data
Pure mathematical model — no external data required. Results can be verified by comparison with published ballistics tables.
Mathematical approach
Write y(t) = h + v₀sinθ · t − ½gt²; solve for time of flight T when y = 0; substitute into x = v₀cosθ · T; differentiate x with respect to θ and set to zero; solve for the optimal θ as a function of h; show that θ → 45° as h → 0.
Inspired by
IB AI HL Paper 3, May 2024
AA SL#10

How well does the sinusoidal model T(m) = a·sin(bm + c) + d fit monthly temperature data for a chosen city, and how do the parameters compare between a tropical and a temperate city?

Data
Meteostat (meteostat.net) or Weather Underground (wunderground.com) — historical monthly mean temperatures for any weather station, free, no registration required.
Mathematical approach
Collect 12-month temperature data for two cities; use GDC sinusoidal regression to fit T(m); compare amplitude a (seasonal range), midline d, and goodness of fit R²; interpret each parameter physically; discuss why the tropical city has a much smaller amplitude.
Inspired by
IB AI HL Paper 3, May 2021
AA SL#11

How does the doubling time of an exponential growth model P(t) = P₀e^(rt) depend on the growth rate r, and how accurate is the financial ‘Rule of 70’ approximation?

Data
Pure mathematical exploration — no external data required. Real-world growth rates (e.g. population, inflation) can be sourced from World Bank data for a grounding example.
Mathematical approach
Set P₀e^(rT) = 2P₀ to derive T = ln2/r; compare with T ≈ 70/(100r); compute the relative error as a function of r; plot both formulas and identify where the approximation fails; extend to the ‘Rule of 72’ and compare accuracy.
Inspired by
IB AA HL Paper 3, May 2023
AA SL#12

For a family of quadratic functions y = ax² + bx + c with fixed discriminant Δ = b² − 4ac = k, how does the distance between the x-intercepts change as a varies?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Use the quadratic formula to express the distance between roots as d = √k / |a|; show d is inversely proportional to |a|; verify graphically on a GDC; determine when d exceeds a chosen threshold and express the answer in terms of k and a.
Inspired by
IB AA HL Paper 3, May 2022
AA SL#13

How many terms of the Maclaurin series for cos(x) are needed to guarantee accuracy within 0.001 for all x in [−π/2, π/2]?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Write cos(x) = Σ (−1)ⁿ x^(2n)/(2n)!; apply the Lagrange error bound |R_n(x)| ≤ |x|^(n+1)/(n+1)! with |x| ≤ π/2; find the minimum n such that (π/2)^(n+1)/(n+1)! < 0.001; verify numerically using a GDC.
Inspired by
IB AA HL Paper 3, May 2021
AA SL#14

For a ball dropped from height h that bounces to a fraction r of its previous height each time, how does the total distance travelled compare to the total time taken, and what happens as r → 1?

Data
Pure mathematical model. The coefficient of restitution r can optionally be verified by filming a real ball bounce with a smartphone and measuring successive peak heights.
Mathematical approach
Sum distance as a geometric series D = h(1 + 2r + 2r² + …) = h(1 + 2r/(1−r)); sum time using T_n = √(2h_n/g) as another geometric series; express total time as a function of r; analyse the ratio D/T as r → 1 and interpret physically.
Inspired by
IB AA HL Paper 3, May 2022
AA SL#15

What are the dimensions of a cylindrical can of fixed volume V that minimise its total surface area, and is the optimal height-to-radius ratio independent of the volume?

Data
Pure mathematical exploration — no external data required. The result can be verified against real-world can dimensions (measure a standard drinks can).
Mathematical approach
Express SA = 2πr² + 2πrh with constraint πr²h = V; substitute h = V/(πr²); differentiate SA with respect to r; solve to get r = (V/2π)^(1/3); confirm h = 2r for all V; compare with actual beverage can dimensions and discuss why manufacturers deviate.
Inspired by
IB AI HL Paper 3, May 2023
AA SL#16

How does the area of a triangle inscribed in a fixed circle change as one vertex moves along the arc, and at which position is the area maximised?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Fix two vertices on a circle of radius R; express area A in terms of the arc angle θ of the third vertex; differentiate and solve; prove that the maximum area triangle inscribed in a circle is equilateral; verify the result using the GDC.
Inspired by
IB AA HL Paper 3, May 2021
AA SL#17

For a savings account with a fixed monthly deposit of £d and annual interest rate r, how does the total value after n years compare between starting deposits early versus late?

Data
Pure mathematical exploration. Real interest rates can be sourced from any savings account provider for a grounding example.
Mathematical approach
Model total value as a geometric series S = d · [(1 + r/12)^(12n) − 1] / (r/12); compute S for n = 5, 10, 20, 30 years; compare starting 5 years earlier vs later; find the break-even crossover year at which the later investor cannot catch up.
Inspired by
IB AA HL Paper 3, May 2024
AA SL#18

What is the relationship between the arithmetic and geometric means of two positive numbers, and how close is the geometric mean to the arithmetic mean as their ratio changes?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Express AM = (a+b)/2 and GM = √(ab); prove AM ≥ GM algebraically; compute the ratio GM/AM as a function of k = b/a; show GM/AM → 1 as k → 1 and GM/AM → 0 as k → ∞; explore when the two means agree to within 1%.
Inspired by
IB AA HL Paper 3, May 2023
AA SL#19

How does the shape of the optimal rectangular enclosure change when the fencing costs are not equal on all four sides, and what cost ratio forces the optimal shape to be a square?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Minimise cost C = 2c₁x + 2c₂y subject to fixed area xy = A; use substitution and differentiation to find optimal x = √(Ac₂/c₁), y = √(Ac₁/c₂); show cost ratio c₁/c₂ = 1 gives a square; explore how the shape elongates as c₁/c₂ varies.
Inspired by
IB AA HL Paper 3, May 2023
AA SL#20

How does the expected waiting time in a geometric probability model compare to the actual waiting times observed when rolling a die repeatedly, and does a chi-squared test confirm the model fits?

Data
Student self-collected: roll a standard die until a six appears and record the number of rolls. Repeat 50–100 times (approximately 30 minutes). No specialist equipment needed.
Mathematical approach
Use E(X) = 1/p = 6 and P(X = k) = (5/6)^(k−1) × (1/6) for the geometric model; compute expected frequencies for k = 1, 2, 3, 4, 5, ≥6; compare with observed counts; apply χ² goodness-of-fit test; discuss whether the independence and constant p assumptions hold.
Inspired by
IB AA HL Paper 3, May 2024

Analysis & Approaches — Higher Level 10 questions

AA HL#1

For the family f(x) = x³ − 3cx + d, how does the number and nature of real roots change with c and d, and what is the equation of the boundary curve in the (c, d) parameter plane?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Find stationary points by solving f’(x) = 0 (at x = ±√c); evaluate f at each stationary point; show the boundary between one and three real roots is 4c³ = d²; use implicit differentiation to find its tangent; classify regions by discriminant sign.
Inspired by
IB AA HL Paper 3, May 2021
AA HL#2

Can mathematical induction prove that ∫₀^∞ xⁿe⁻ˣ dx = n!, and what does this reveal about the connection between integration and the factorial function?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Compute A₀ = ∫₀^∞ e^(−x) dx = 1 as the base case; use integration by parts to derive A_n = n · A_{n−1}; write a rigorous proof by induction that A_n = n!; connect to the Gamma function Γ(n + 1) = n!; verify numerically for n = 4 and 5.
Inspired by
IB AA HL Paper 3, May 2023
AA HL#3

Can a single closed-form formula unify triangular, square, pentagonal, and general r-sided polygonal numbers, and can it be proved by mathematical induction?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Show the n-th r-gonal number is the sum of an arithmetic sequence with first term 1 and common difference r − 2; derive P(r, n) = n[(r − 2)n − (r − 4)]/2; prove by induction; verify for r = 3, 4, 5 against known sequences.
Inspired by
IB AA HL Paper 3, May 2022
AA HL#4

For a family of cubic polynomials sharing complex conjugate roots a ± bi, how does the tangent at the turning point always pass through the real root, regardless of b?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Factor g(x) = (x − r)(x² − 2ax + a² + b²); compute g’(x) and evaluate the tangent at x = a; show algebraically that the tangent passes through (r, 0) for any b; explore how the distance from (a, g(a)) to the x-axis changes as b varies.
Inspired by
IB AA HL Paper 3, May 2022
AA HL#5

How does the maximum product of n positive reals with fixed sum S depend on n, and how do the AM-GM inequality and a calculus approach provide complementary proofs?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Apply AM-GM: (x₁…xⁿ)^(1/n) ≤ S/n with equality iff all x_i = S/n; also maximise g(x) = x·ln(S/x) using differentiation to find the continuous extension; compare both proofs; explore M_n(S) = (S/n)ⁿ and its limit as n → ∞.
Inspired by
IB AA HL Paper 3, May 2023
AA HL#6

For the quartic y = x⁴ − mx³ + nx, how does the ratio (x_B − x_A)/(x_C − x_B) — where B and C are points of inflexion and A is the further intersection of line BC with the curve — depend on m?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Find inflexion points by solving y’’ = 12x² − 6mx = 0, giving x_B = 0 and x_C = m/2; find the equation of line BC; substitute back to locate A; compute the ratio; prove it equals (√5 − 1)/2 (the reciprocal of the golden ratio, independent of m and n).
Inspired by
IB AA HL Paper 3, May 2024
AA HL#7

How do the expected value and variance of a standard geometric distribution compare with those of a ‘resetting escalating probability’ distribution, and which model gives a more consistent player experience?

Data
Pure mathematical exploration — no external data required. This question is inspired by probability mechanics in video games and online reward systems.
Mathematical approach
Compute E(X) = 1/p and Var(X) = (1−p)/p² for the geometric model; for the escalating model (p increases by 0.2 after each failure, resets after success) compute P(Y = k) explicitly for k = 1,…,5; find E(Y) and Var(Y); compare variances for equal means.
Inspired by
IB AA HL Paper 3, May 2024
AA HL#8

How many terms of the Maclaurin series for e^x are needed to approximate e to within 10⁻⁸, and how does the Lagrange remainder bound compare to the actual numerical error?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Write e^x = Σ xⁿ/n!; apply |R_n(1)| ≤ e/(n+1)! as the Lagrange bound; find minimum n; compute actual errors for n = 5, 10, 15 by direct comparison; plot log₁₀(error) vs n and comment on the rate of convergence.
Inspired by
IB AA HL Paper 3, May 2021
AA HL#9

For the logistic differential equation dP/dt = kP(L − P), how does the time required to reach half the carrying capacity depend on the initial population P₀ and growth rate k?

Data
Pure mathematical exploration — no external data required. The result can optionally be validated against a real population dataset (e.g. World Bank historical population data).
Mathematical approach
Separate variables and integrate to find P(t) = L/(1 + Ce^(−kLt)) with C = (L − P₀)/P₀; set P(t*) = L/2 to get t* = ln(C)/(kL); show t* depends logarithmically on C; compare scenarios with P₀ = 0.01L and P₀ = 0.1L.
Inspired by
IB AI HL Paper 3, May 2022
AA HL#10

How does the complex-root structure of a polynomial with real coefficients constrain the shape of its real graph, and can the polynomial be reconstructed from its graph alone?

Data
Pure mathematical exploration — no external data required.
Mathematical approach
Apply the conjugate root theorem; explore degree-4 polynomials with different root configurations (four real, two real + two complex, all complex); use Vièta’s formulas to relate roots to coefficients; reconstruct the polynomial from observed graph features (roots, turning points, inflexions).
Inspired by
IB AA HL Paper 3, May 2022

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