IB Maths AA IA Ideas: 60 High Scoring Exploration Topics for SL and HL

For a cylinder of fixed volume, which height-to-radius ratio minimises surface area, and why do real cans ignore the answer?

• Apply the volume constraint
• Differentiate S(r)
• Weight the material costs
• Measure real cans

Mathematics. Substitution to reduce two variables to one, differentiation of 2πr² + 2V/r, the second derivative test.

Push further. Let the ends cost k times the wall per unit area, show the optimum becomes h/r = 2k, then measure a dozen real cans and estimate the k they imply.

Failure mode. h = 2r is a two-line result. Without the cost-weighted generalisation there is not enough mathematics here to sustain an exploration.

Which dimensions minimise the material in an open-topped box of fixed volume, and how does the answer move when the base costs more than the sides?

• Model the surface area
• Eliminate a variable
• Minimise
• Re-weight and re-solve

Mathematics. Algebraic substitution, differentiation of a function with a reciprocal term, justification of the minimum.

Failure mode. The cut-squares-from-a-sheet version is the same problem in a different costume. Doing both thinly is weaker than doing one of them completely, including the cost-weighted case.

Given different running and swimming speeds, at which point along the shoreline should a lifeguard enter the water to reach a swimmer soonest?

• Build the total-time function
• Differentiate
• Solve numerically
• Interpret the condition

Mathematics. Distance formula, chain rule applied to square roots, optimisation with a variable confined to an interval.

Push further. Show that the optimum satisfies sinθ₁/v₁ = sinθ₂/v₂ and recognise Snell’s law falling out of a rescue problem.

Failure mode. The stationary equation has no tidy closed form. Say so and solve it numerically, rather than quietly rounding your way to a false exact answer.

What price maximises revenue for a school event, given a demand model estimated from a survey of the people who would actually buy tickets?

• Survey demand
• Justify a demand model
• Derive R(p)
• Test sensitivity

Mathematics. Linear or exponential demand functions, the revenue function R(p) = p D(p), differentiation, interpretation of the stationary point.

Failure mode. AA gives no credit for the regression itself. The marks sit in deriving R(p), defending the shape of D(p), and showing how far the optimal price moves when you change that shape.

For a fixed 400 m perimeter, how far do a minimum bend radius and a minimum straight move the maximum enclosed area away from the circle?

• Find the unconstrained optimum
• Note it is degenerate
• Impose constraints
• Re-optimise

Mathematics. Perimeter constraint, area as a function of one variable, constrained optimisation, comparison with the official track geometry.

Failure mode. Unconstrained, the answer is simply a circle with no straights at all. The exploration only exists once realistic constraints are imposed — which is precisely the interesting mathematics, provided you say why.

From a fixed area of sheet material, which conical container has the greatest capacity?

• Fix the material
• Express V in terms of r
• Maximise
• Compare with real containers

Mathematics. Slant height and curved surface area, V = ⅓πr²h, substitution, differentiation, verification of the maximum.

Failure mode. Maximising volume divided by surface area with nothing fixed has no answer at all: that ratio has the dimensions of a length and grows without bound as you scale the shape up. Fix the material or fix the volume before you differentiate anything.

How does the range-maximising launch angle change when a projectile is released above the level at which it lands?

• Derive the trajectory
• Express range
• Optimise over the angle
• Tabulate against height

Mathematics. Parametric motion, trigonometry, differentiation with respect to an angle, limiting behaviour.

Push further. Obtain sinθ = 1/√(2 + 2gh/u²) and examine what happens as h → 0 and as h grows large.

Failure mode. Forty-five degrees is a consequence of assuming equal launch and landing heights. An exploration that verifies 45° has verified its own assumption instead of testing it.

Which release angle leaves the largest margin for error while still sending the ball through the hoop?

• Define the margin
• Model the trajectory
• Search over angles
• Compare with players

Mathematics. Quadratic trajectories, the geometry of ball and hoop diameters, entry angle, optimisation of a derived objective.

Failure mode. “Best angle” means nothing until you state what is being optimised. Minimum release speed and maximum angular tolerance give genuinely different answers, and choosing between them is the exploration.

At what distance from a cinema screen is the vertical viewing angle greatest?

• Model the geometry
• Differentiate arctan
• Solve
• Test on a real auditorium

Mathematics. θ = arctan((h+d)/x) − arctan(d/x), the derivative of arctan, optimisation, verification against measured dimensions.

Failure mode. This is labelled SL almost everywhere online, and it should not be. Both routes to it are HL content: differentiating arctan, or using the compound-angle identity for tangent. At SL it collapses into reading a maximum off a graph.

Which fixed tilt angle maximises the annual solar energy collected at one latitude?

• Model the incidence angle
• Find the daily optimum
• Aggregate over a year
• Compare with measurement

Mathematics. Solar declination through the year, the cosine of the incidence angle, integration or summation over the year, optimisation.

Data. Measured irradiance by latitude is published free through NASA POWER, which is enough to test a geometric prediction properly.

Failure mode. Clear-sky geometry ignores cloud, and cloudy climates shift the measured optimum away from the geometric one. That gap is the reflection, not an embarrassment.

How does the error in different numerical integration methods fall as the number of intervals increases?

• Choose a well-behaved integral
• Compute errors
• Plot log error against log n
• Identify the order

Mathematics. Riemann sums, midpoint and trapezium rules, π = 4∫₀¹ dx/(1+x²), logarithmic error analysis.

Push further. Derive the Leibniz series from the Maclaurin expansion of arctan and compare how slowly it converges against the numerical methods.

Failure mode. The popular choice, 4∫₀¹√(1−x²) dx, is the worst possible one. Its derivative is infinite at x = 1, so the trapezium error decays like h^1.5 rather than h² and your convergence study will appear to contradict the theory. Use the arctan integral.

Why does the Newton–Raphson iteration roughly double the number of correct decimal places at each step, and when does it fail?

• Derive the iteration
• Expand the error
• Measure convergence
• Construct failures

Mathematics. Tangent-line iteration, Taylor expansion of the error term, the relation eₙ₊₁ ≈ (f″/2f′)eₙ², fixed points.

Push further. Find a starting value that cycles forever, and one where the method converges to the wrong root; explain both from the derivative.

Failure mode. The method is not on the syllabus, so the derivation must be yours. An exploration that only iterates and tabulates has done arithmetic, not analysis.

How long is a chosen curved road, and how does the answer depend on how finely it is measured?

• Approximate with chords
• Take the limit
• Derive the integral
• Vary the resolution

Mathematics. Deriving ∫√(1+(f′)²) dx from Pythagoras, curve fitting to mapped coordinates, numerical integration, parametric alternatives.

Failure mode. Arc length appears nowhere in the AA guide, at either level, so quoting the formula wastes the best part of the topic: derive it. The coastline version of this question leads towards fractal dimension, which belongs to a different course entirely.

How accurately can the capacity of an irregular container be predicted from a function fitted to its profile?

• Measure the profile
• Fit and justify a function
• Integrate
• Fill it and compare

Mathematics. Curve fitting to measured points, V = π∫y² dx, comparison of competing fitted functions, percentage error.

Failure mode. Solids of revolution are HL content in AA; SL integration reaches the area under a curve and stops. An SL student can build a slice-by-slice numerical model instead, but must present it as exactly that rather than as a volume integral.

To what extent does an exponential model describe a cooling drink, and how does a lid change the parameters?

• Record temperatures
• Linearise with logarithms
• Estimate k
• Compare conditions

Mathematics. Exponential functions, logarithmic linearisation, parameter estimation, rates of change.

Push further. Derive the model by separating dT/dt = −k(T − Tₐ), which converts a school experiment into a genuine HL exploration.

Failure mode. Assuming the ambient temperature rather than fitting it is the classic error. Expect the log plot for an open cup to curve, because evaporation cools it by a second mechanism the model does not contain.

How does the shape of a container determine the time for liquid to drain through a small opening?

• Set up the differential equation
• Separate variables
• Integrate
• Test against a real container

Mathematics. Torricelli’s law v = √(2gh), the equation A(h) dh/dt = −a√(2gh), separable first-order differential equations, comparison across profiles.

Failure mode. The ideal model under-predicts the drain time by roughly half again, because the emerging jet contracts to about 60% of the hole area. Predicting that discrepancy before you measure it is a far stronger exploration than discovering it afterwards and calling it experimental error.

Which curve describes a freely hanging chain, and which describes the cable of a loaded suspension bridge?

• Derive the parabola from deck load
• Derive the catenary from self-weight
• Fit both to a photograph
• Quantify the difference

Mathematics. Force balance leading to a differential equation, y = a cosh(x/a) written explicitly through exponentials, curve fitting, residual comparison.

Failure mode. Over a typical span the two curves differ by less than the error in reading points off a photograph, so “which fits better?” returns a null result and an exploration with nothing to say. The derivations are the content; the fit is only a check.

How much faster is a cycloidal descent than a straight line between the same two points?

• Convert energy to speed
• Build the time integral
• Evaluate for three paths
• Compare and interpret

Mathematics. Conservation of energy giving v = √(2gy), the descent time ∫ds/v, parametric equations of the cycloid, numerical integration.

Failure mode. The integrand is singular at the release point, so naive numerical integration converges painfully slowly or not at all; use a substitution and justify it. Do not attempt the calculus of variations — comparing three specified paths is the version that works.

Over what range of angles does sin x ≈ x stay within a stated relative error, and what does that mean for a pendulum?

• Define the error precisely
• Tabulate
• Locate the threshold
• Apply to a real system

Mathematics. Relative and absolute error functions, numerical solution, graphical comparison, application to small-angle pendulum motion.

Push further. Use the Maclaurin series to show the relative error behaves like x²/6, predicting the 1% threshold at about 0.245 radians — roughly 14 degrees — before you check it numerically.

Failure mode. State whether the error is absolute or relative. The two give different thresholds, and most versions of this topic never say which they mean.

How many terms are required to hold a function within a chosen error across a given interval, and how does that depend on the function?

• Generate the series
• Tabulate error against terms
• Find the usable interval
• Compare functions

Mathematics. Maclaurin expansions of eˣ, sin x and cos x, partial sums, behaviour of the remainder, alternating-series reasoning.

Failure mode. The Lagrange error bound is not in the AA HL guide. Either derive a bound you can justify — the alternating-series argument works cleanly for sine and cosine — or state openly that your error study is empirical.

Why does the ratio of consecutive Fibonacci numbers converge to the golden ratio, and how quickly does it get there?

• Compute the ratios
• Derive x = 1 + 1/x
• Justify the limit
• Measure the rate

Mathematics. Recurrence relations, fixed points, limits, geometric decay of the error.

Push further. Derive Binet’s formula and show the error shrinks like (−1/φ)ⁿ, which explains the alternating overshoot in your table.

Failure mode. Measuring golden ratios in faces, shells and buildings is not mathematics and reliably scores badly. Keep the entire exploration inside the sequence.

How many terms of a geometric series are needed for a chosen accuracy, and how does that number depend on the common ratio?

• Derive the partial sum
• Express the error
• Solve for n
• Tabulate against r

Mathematics. Partial sums, the infinite sum for |r| < 1, logarithms to solve an inequality for n.

Failure mode. This needs a real question attached to it — a repeating decimal, a bouncing ball, a drug that accumulates — or it is textbook work with a title on top.

How does increasing the compounding frequency lead mathematically to continuous growth?

• Tabulate (1 + r/n)ⁿ
• Estimate the limit
• Derive it using logarithms
• Compare with eʳ

Mathematics. Sequences, limits, logarithms, exponential functions, rate of approach to the limit.

Failure mode. Comparing savings accounts is not the exploration. The limit, and how quickly it is approached, is.

How does the interval between doses determine the peak and trough concentrations that a repeated-dose model approaches?

• Model a single dose
• Sum the geometric series
• Find the steady state
• Vary the interval

Mathematics. Exponential decay, half-life, infinite geometric series, the limiting peak D/(1 − e⁻ᵏᵗ), limits.

Push further. Rebuild the model as a differential equation with periodic input and compare the two descriptions.

Failure mode. Half-life alone is a single calculation. The geometric series is what makes this an exploration. Keep it a modelling exercise and avoid any claim that reads as clinical advice.

Which closed forms and identities connect triangular, square and general k-gonal numbers?

• Generate the sequences
• Conjecture a formula
• Derive it
• Generalise to k sides

Mathematics. Differences, deriving closed forms from a pattern, deductive proof, algebraic identities.

Push further. Prove your general relationship by induction, which is HL content and exactly what this topic is for.

Failure mode. Listing patterns is not proving them. At SL a laid-out deductive argument is expected; at HL, induction. Neither is optional.

Why does the hockey-stick identity hold, and what structure appears when the triangle is reduced modulo a prime?

• Find an identity
• Prove it two ways
• Reduce modulo 2
• Describe the structure

Mathematics. Binomial coefficients, combinatorial and algebraic proof of the same identity, modular arithmetic, self-similarity.

Failure mode. The modulo-2 pattern is easy to display and hard to explain. If you show the picture, you must argue why it appears; otherwise the exploration is decoration.

What is the minimum number of moves for n discs, and how does the answer change with a fourth peg?

• Play and record
• Establish Tₙ = 2Tₙ₋₁ + 1
• Find the closed form
• Extend to four pegs

Mathematics. Recurrence relations, the closed form 2ⁿ − 1, deductive argument for the lower bound.

Push further. Prove the closed form by induction, then investigate the Frame–Stewart pattern for four pegs.

Failure mode. A formula guessed from a table is a conjecture, not a result. Induction is HL content; at SL, verify the recurrence carefully and argue why no shorter sequence of moves exists.

The classical proof shows √2 is irrational — but exactly how far does that argument generalise?

• Reconstruct the classical proof
• Find where it fails for √4
• Generalise to √n
• Test it on log₂3

Mathematics. Proof by contradiction, parity arguments, unique factorisation, generalisation of a proof technique.

Failure mode. The parity argument is not the general argument, and noticing precisely which step must be replaced by the fundamental theorem of arithmetic is the whole exploration. Reciting the standard proof alone is bookwork.

How accurately does x/ln x estimate the number of primes below x, and does the error behave predictably?

• Generate π(x)
• Compare estimators
• Analyse relative error
• Extrapolate cautiously

Mathematics. The prime-counting function, logarithms, relative error, comparison with the logarithmic integral.

Failure mode. Over ranges you can actually compute, x/ln x is about 8% low at one million and improves very slowly. That slow convergence is the finding — report it rather than treating it as a mistake, and do not attempt to prove the prime number theorem.

Why do continued-fraction convergents give the best rational approximations, and why is the golden ratio the hardest number to approximate?

• Expand a number
• Generate convergents
• Measure the error
• Compare with rival fractions

Mathematics. Continued fractions, the recurrence for numerators and denominators, error bounds, limits.

Failure mode. Off-syllabus, so the convergent recurrence has to be derived rather than quoted. Use 355/113 for π as your test case: it is accurate to seven decimal places, and explaining why is the exploration.

For a fixed perimeter, how does the area of a regular n-gon approach that of a circle as n increases?

• Derive the area formula
• Tabulate
• Take the limit
• Interpret

Mathematics. Area of a regular n-gon as (P²/4n) cot(π/n), radian trigonometry, limits.

Push further. Evaluate the limit analytically using the small-angle behaviour of the tangent, rather than reading it off a table.

Failure mode. Fixing the perimeter and fixing the circumradius give opposite stories. State which is held constant in the first line, or the whole comparison is meaningless.

How quickly do inscribed and circumscribed polygon perimeters close in on π?

• Derive the doubling recurrence
• Iterate
• Measure the bounds
• Investigate breakdown

Mathematics. Half-angle relationships, recurrence relations, upper and lower bounds, convergence rate.

Push further. This is the best finding available in the whole section: the naive recurrence subtracts two nearly equal numbers, so in ordinary floating-point arithmetic the estimate stops improving and then gets steadily worse. Rearranging the recurrence algebraically removes the cancellation and fixes it.

Failure mode. Reproducing Archimedes and stopping is a history project. The convergence rate, or the arithmetic breakdown above, is what makes it an exploration.

How does the angle sum of a spherical triangle depend on its area?

• Find the area of a lune
• Combine three lunes
• Derive the excess formula
• Test on real coordinates

Mathematics. Areas on a sphere, proportional reasoning, Girard’s theorem, application to three cities’ latitudes and longitudes.

Failure mode. Keep to the lune argument, which is genuinely accessible and complete. Full spherical trigonometry is a different exploration and a much longer one.

How much shorter is the great-circle route than the path that looks straight on a Mercator map?

• Convert coordinates to vectors
• Take the scalar product
• Find the central angle
• Compare with the map

Mathematics. Position vectors on a sphere, the scalar product, arc length rθ, comparison with a projected path.

Failure mode. Vectors are HL-only content in AA, and the spherical cosine rule is off-syllabus. Deriving the distance through the scalar product keeps the exploration inside the course; quoting the haversine formula from a website does not.

How does the scale distortion of the Mercator projection grow with latitude?

• Derive the projection
• Differentiate
• Quantify area inflation
• Apply to two countries

Mathematics. The integral ∫sec φ dφ giving ln|sec φ + tan φ|, differentiation, area scaling by sec²φ.

Failure mode. The Greenland-versus-Africa comparison is the hook, not the exploration. The integration and the derivation of the scale factor are what earn Criterion E.

Why does every ray parallel to the axis of a parabola reflect through the focus?

• Derive from focus and directrix
• Find the tangent
• Show the angles are equal
• Apply to a dish

Mathematics. The focus–directrix definition, differentiation to obtain the tangent gradient, angle comparison, coordinate geometry.

Push further. Complete the angle argument using the compound-angle identity for tangent instead of a similar-triangles shortcut.

Failure mode. This is a proof, so it has to be complete. “Satellite dishes are parabolic” is context; the equality of the two angles is the content.

What is the shortest path between two points that must touch a wall on the way, and how does the answer extend to two walls?

• Reflect one point
• Draw the straight line
• Verify with calculus
• Extend to two cushions

Mathematics. Reflection in a line, coordinate geometry, the triangle inequality, confirmation by differentiation.

Failure mode. The reflection trick makes the first answer immediate, which leaves nothing to explore unless you go further: two walls, a corner shot, or a proper argument that the reflected path really is the minimum.

For two objects moving on straight courses at constant speeds, when is the separation least, and how small does it get?

• Write both paths as vector equations
• Form the separation vector
• Minimise its square
• Compare with the skew-line distance

Mathematics. Vector equations of lines, kinematics with vectors, minimisation of |d(t)|², the scalar product, the shortest distance between skew lines.

Failure mode. Minimise the square of the distance, not the distance; the square root adds algebra and nothing else. And keep the two questions separate: the closest approach in time and the shortest distance between the tracks as infinite lines have different answers.

Under inversion in a circle, which lines and circles keep their type, and which change?

• Define the transformation
• Transform a general line
• Transform a general circle
• Classify the cases

Mathematics. The relation OP·OP′ = r², coordinate algebra, transformation of equations, classification.

Failure mode. Entirely off-syllabus, so every claim needs its algebra shown. A gallery of before-and-after pictures without the transformed equations earns very little under Criterion E.

Is the circle the only shape that rolls smoothly, and does the perimeter of a constant-width curve depend on its shape?

• Construct a Reuleaux triangle
• Compute its perimeter
• Compute its area
• Compare with the circle

Mathematics. Circular arcs and radian measure, perimeter and area by decomposition, comparison of a family of shapes, Barbier’s theorem for Reuleaux polygons.

Failure mode. That every constant-width curve of width w has perimeter πw is startling, and provable for Reuleaux polygons but not in general at this level. Bound the claim you make to the case you can actually prove.

How many people are needed for an even chance of a shared birthday, and does the unevenness of real birth dates help or hinder a match?

• Derive p(n)
• Locate the 50% threshold
• Model uneven birth rates
• Compare

Mathematics. Complementary probability, products of probabilities, logarithms for an approximate threshold, weighted probability models.

Push further. Argue the striking result that any departure from uniform birth rates increases the probability of a coincidence, then illustrate it with published monthly birth data.

Failure mode. Do not reach for a chi-squared test to compare model with data: AA contains no hypothesis testing at all, and importing one signals an Applications and Interpretation exploration.

How many shortest routes cross a grid, and how do closed streets and compulsory checkpoints change the count?

• Count unrestricted paths
• Force a checkpoint
• Subtract blocked routes
• Generalise

Mathematics. Binomial coefficients as choices of direction, multiplication principle, complementary counting, recursion.

Push further. Derive a recursion for the blocked grid and connect it to Pascal’s triangle, or to Catalan-type counting when a diagonal may not be crossed.

Failure mode. The plain count is one line of work. The exploration lives in the obstacles, and specifically in arguing that your subtraction has not removed the same route twice.

How does the advantage of switching change when there are n doors and the host opens k of them?

• Solve the three-door case
• State the host’s rules
• Generalise to n and k
• Confirm by simulation

Mathematics. Conditional probability, tree diagrams, generalisation of a probability argument.

Failure mode. Retelling the paradox scores nothing, and the answer depends entirely on rules most students never state: that the host always opens a door, and never the winning one. Write the rules down first.

What proportion of candidates should be rejected before choosing, and how well does the rule work for small numbers?

• Derive P(n, k)
• Optimise k
• Let n grow
• Test small cases

Mathematics. Probability sums, harmonic sums, optimisation over an integer, the limit approaching 1/e.

Failure mode. An unexplained 37% is worth nothing. The derivation is the exploration — and the small-n results converge on the limit remarkably quickly, which is worth reporting in its own right.

After n steps, how far from the origin should a one-dimensional random walk be expected to be?

• Find the expected position
• Find the variance
• Predict typical distance
• Simulate and compare

Mathematics. Expectation and variance, the binomial distribution, independence, simulation as verification.

Push further. Derive the probability of returning to the origin using central binomial coefficients.

Failure mode. The expected absolute distance needs Stirling’s approximation, which is off-syllabus. Either derive it honestly or work with the standard deviation √n instead and say why.

For a small game of chance, how large is the first player’s advantage, and what rule change would remove it?

• Define the states
• Write first-step equations
• Solve simultaneously
• Verify by simulation

Mathematics. Conditional probability, first-step analysis, simultaneous equations, expected value.

Failure mode. Transition matrices and Markov chains are Applications and Interpretation HL content; AA has no matrices at any level. First-step analysis reaches the same answers with AA algebra. Choose a game small enough to solve exactly — full Monopoly is not.

How many perfect riffle shuffles return a deck of cards to its original order, and how does that depend on the size of the deck?

• Model the shuffle as a map
• Reduce it modulo n − 1
• Compute the order
• Vary the deck size

Mathematics. Permutations, the map x → 2x modulo (n − 1), modular arithmetic, the order of an element, patterns across deck sizes.

Failure mode. The popular version of this topic asks how many shuffles make a deck “random enough”, which has no definition you can work with at this level. This version does, and the fact that 52 cards return after eight out-shuffles is something you can prove.

What mixture of left and right makes a penalty taker impossible to exploit, and do real players follow it?

• Build the payoff matrix
• Set expected values equal
• Solve for the mixture
• Compare with real frequencies

Mathematics. Expected value, simultaneous equations, indifference conditions, mixed strategies.

Failure mode. Real kickers deviate from equilibrium. Explaining why — stronger foot, fatigue, reading the keeper — is better reflection than insisting the model is exact.

How efficient is Buffon’s needle as a method of estimating π?

• Derive the crossing probability
• Run the experiment
• Invert to estimate π
• Analyse the convergence

Mathematics. Geometric probability, integration over an angle to obtain p = 2ℓ/(πd), expectation and variance of a proportion, convergence.

Failure mode. The estimate converges like 1/√N, so three correct decimal places need on the order of a million trials. Reporting that inefficiency honestly is the conclusion; hiding it behind a lucky run of 100 drops is not.

What is the least number of rounds in which every team can play every other exactly once, and how is such a schedule built?

• Count the fixtures
• Bound the rounds
• Construct by the circle method
• Prove it works

Mathematics. Combinations, parity and counting bounds, modular arithmetic, a constructive existence proof.

Failure mode. Graph theory and adjacency matrices belong to Applications and Interpretation HL. The circle method with a modular-arithmetic proof reaches the same result and is entirely at home in AA.

How quickly does a Fourier series converge to a square wave, and what happens at the jump?

• Derive the coefficients
• Sum partial series
• Measure the error
• Examine the discontinuity

Mathematics. Integration by parts, orthogonality integrals, coefficient formulae, partial sums, error measurement.

Failure mode. The overshoot at the jump does not shrink as you add terms — it narrows but keeps roughly the same height. Expect that and explain it, rather than reporting it as a computational error.

Why does the equation x″ = −ω²x force sinusoidal motion, and how well does a real oscillator obey it?

• Write the equation
• Reduce with v dv/dx
• Separate twice
• Compare with measurement

Mathematics. The substitution a = v dv/dx, separable first-order equations, energy conservation, recovery of x(t).

Failure mode. AA HL covers first-order differential equations only, so the reduction to first order is the honest route. Substituting a sine and observing that it works demonstrates far less than deriving it, and examiners can tell the difference immediately.

What structure do the solutions of zⁿ = 1 have, and what can be deduced from their sum?

• Solve in modulus–argument form
• Plot on the Argand diagram
• Prove the sum is zero
• Extract an exact value

Mathematics. Complex numbers, De Moivre’s theorem, geometric series in the complex plane, regular polygons.

Push further. Use the fifth roots of unity to derive an exact value for cos(2π/5), and hence construct a regular pentagon.

Failure mode. Plotting the roots is a diagram. Proving the sum vanishes and using it to extract an exact cosine is an exploration.

Under the iteration z → z² + c, which values remain bounded, and how can that be decided rather than observed?

• Iterate by hand
• Conjecture a bound
• Prove the escape criterion
• Map a slice of the plane

Mathematics. Complex modulus, iteration, inequalities, proof that once |z| exceeds 2 the sequence diverges.

Failure mode. The famous image is not the mathematics, and an exploration that produces one earns almost nothing. The provable core is the escape criterion, and that proof is short enough to do properly.

Under what conditions do exponential and logistic models give materially different predictions?

• Solve both equations
• Compare the solutions
• Fit to data
• Examine long-run behaviour

Mathematics. Separable differential equations, partial fractions to integrate the logistic equation, limits, parameter estimation.

Data. Long, clean population and epidemiological series are published free by Our World in Data and the World Bank.

Failure mode. Early in the growth the two curves are almost identical, so a short data window cannot distinguish them at all. Choosing a window that can is part of the design, and saying why is part of the reflection.

In a simplified SIR model, how does the transmission parameter determine the largest proportion infected at any one time?

• Eliminate time via dI/dS
• Integrate
• Derive the peak
• Confirm with Euler’s method

Mathematics. Coupled differential equations, elimination of the independent variable, integration giving I + S − (γ/β) ln S constant, Euler’s method as verification.

Push further. Obtain the closed form for the peak, I_max/N = 1 − (1 + ln R₀)/R₀, and test its sensitivity to R₀.

Failure mode. Almost every version of this topic only plots numerical output. Eliminating time gives an exact result for the peak, and that piece of analysis is what separates a strong SIR exploration from a plotted one. Keep it a study of the model, not a medical prediction.

At what point does a discrete growth recurrence stop agreeing with its continuous counterpart?

• Build the recurrence
• Derive the continuous model
• Compare predictions
• Increase the growth rate

Mathematics. Recurrence relations, exponential functions, limits, error growth over many steps.

Push further. Iterate the logistic recurrence and watch the stable fixed point lose stability as the growth parameter rises.

Failure mode. For slow growth the two models agree so closely that the comparison has nothing in it. Push the growth rate until they visibly separate, and explain what causes the separation.

How does the step size control the error in Euler’s method, and why does the relationship take the form it does?

• Solve an equation exactly
• Run Euler’s method
• Halve h repeatedly
• Identify the order

Mathematics. Euler’s method, exact solution of a separable equation for comparison, local against global error, error scaling.

Failure mode. Choose an equation you can solve exactly, or you have nothing to measure the error against. Global error should roughly halve when the step halves; anything else usually means an arithmetic slip rather than a discovery.

How can a Maclaurin series be built directly from a differential equation that has no closed-form solution, and where does it stop being useful?

• Differentiate the equation repeatedly
• Assemble the coefficients
• Compare with a numerical solution
• Find where it breaks down

Mathematics. Repeated implicit differentiation, Maclaurin coefficients from derivatives at a point, comparison with Euler’s method.

Failure mode. This technique is explicitly in the HL guide and almost nobody uses it, which makes it a strong choice. But the series is only accurate near the expansion point — showing exactly where it fails is the most valuable part of the exploration, not a flaw in it.

How can polynomial sections of track be joined so that position, gradient and curvature all pass smoothly across the join?

• Choose the sections
• Impose the conditions
• Solve for the coefficients
• Test the curvature

Mathematics. Piecewise polynomials, simultaneous equations from f, f′ and f″ conditions, second derivatives, design constraints.

Failure mode. Matching gradients is easy and matching second derivatives is what actually removes the jolt — and because the gradients already agree at the join, matching f″ there is exactly curvature continuity. Say which condition does what, or the exploration reads as algebra without purpose.