IB DP Maths AA · Exam special · SL & HL · checked against the current AA formula booklets · ~8 min read
IB Maths AA Formula Booklet: What Is Given and What to Memorise
What the IB Maths AA formula booklet (SL and HL) contains, topic by topic, and the formulas, values and facts it leaves out that you must memorise, plus how to use the booklet quickly in the exam.
What the booklet containsSL must-memorise listHL must-memorise listExact valuesCalculus rulesStatistics factsUsing the booklet fast
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againHANDY worth memorisingTest yourself at the end
In 30 seconds
- You get a clean copy of the formula booklet in every AA exam, SL and HL. It's about 10 pages and is organised by syllabus topic.
- It doesn't contain everything you need. Exact trig values, the second derivative test, the rules for transformations and many statistical facts are missing.
- Examiners say students lose marks by not knowing what's in it, or by copying a formula without writing it in the context of the question.
By the end you can
- Know what the AA booklet gives you, topic by topic.
- Memorise the results it leaves out.
- Find any formula in seconds during the exam.
1What the booklet contains
AA SL booklet
| Topic | Given |
|---|---|
| Number and algebra | Arithmetic and geometric sequences and series (including the infinite sum), compound interest, log definition and laws, change of base, binomial theorem, ⁿCᵣ |
| Functions | Straight-line forms and gradient, axis of symmetry, quadratic formula, discriminant Δ = b² − 4ac |
| Geometry and trigonometry | Areas and volumes (prisms, pyramids, cones, spheres, cylinders), 3D distance and midpoint, sine and cosine rules, ½ab sin C, arc length and sector area (radians), tan θ = sin θ/cos θ, Pythagorean identity, double-angle identities for sin and cos |
| Statistics and probability | IQR, mean of a frequency table, probability rules, conditional probability, independence, E(X), binomial mean and variance, z = (x − μ)/σ |
| Calculus | Derivatives of xⁿ, sin x, cos x, eˣ, ln x; chain, product and quotient rules; ∫xⁿ dx; integrals of 1/x, sin x, cos x, eˣ; area under a curve; kinematics (a = dv/dt, distance and displacement integrals) |
Extra in the AA HL booklet
| Topic | Given (HL only) |
|---|---|
| Number and algebra | Permutations ⁿPᵣ, the extended binomial theorem, complex numbers in Cartesian, polar and Euler form, De Moivre's theorem |
| Functions | Sum and product of the roots of a polynomial |
| Geometry and trigonometry | Reciprocal trig functions and their Pythagorean identities, compound-angle identities, tan 2θ, vector magnitude, scalar and vector products, angle between vectors, vector, parametric and Cartesian equations of lines and planes, area of a parallelogram ǀv × wǀ |
| Statistics and probability | Bayes' theorem (two and three events), variance and standard deviation, E(aX + b) and Var(aX + b), continuous random variables (mean, variance) |
| Calculus | First principles, derivatives of tan, sec, cosec, cot, aˣ, logₐx, arcsin, arccos, arctan; integrals of aˣ, 1/(a² + x²), 1/√(a² − x²); integration by parts; area with respect to y; volumes of revolution; Euler's method; integrating factor; Maclaurin series of eˣ, ln(1 + x), sin x, cos x, arctan x |
2What you must memorise: SL and HL
Algebra and functions
- Index laws: aman = am+n, (am)n = amn, a−n = 1/an, a1/n = ⁿ√a, a⁰ = 1.
- Special logs: logₐ1 = 0, logₐa = 1, ln e = 1, eln x = x.
- What Δ tells you: Δ > 0 two real roots; Δ = 0 one repeated root (the curve touches the axis, or a line is a tangent); Δ < 0 no real roots.
- Vertex form: y = a(x − h)² + k has vertex (h, k).
- Parallel lines have equal gradients; perpendicular gradients multiply to −1.
- Inverse functions: swap x and y, rearrange; the domain of f⁻¹ is the range of f; the graph reflects in y = x.
- Transformations: f(x) + a (up a); f(x − a) (right a); af(x) (vertical stretch, factor a); f(ax) (horizontal stretch, factor 1/a); −f(x) (reflect in x-axis); f(−x) (reflect in y-axis).
- Rational functions (x + b)/(cx + d): vertical asymptote x = −d/c, horizontal asymptote y = 1/c.
Trigonometry
- Exact values of sin, cos and tan at 0, π/6, π/4, π/3, π/2, and the unit-circle signs (CAST). See the Paper 1 non-calculator guide.
- 180° = π radians. The booklet's arc and sector formulas need radians.
- Graphs: y = a sin(b(x − c)) + d has amplitude ǀaǀ, period 2π/b, principal axis y = d. The period of tan(bx) is π/b, not 2π/b.
- The ambiguous case: the sine rule can give two triangles when you are finding an angle.
Statistics and probability
- Standard deviation (SL): not given; use the GDC.
- Outliers: below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
- Normal distribution: symmetric about μ; about 68% within 1σ, 95% within 2σ, 99.7% within 3σ.
- Discrete distribution: the probabilities add up to 1.
- Regression: use the line of y on x to predict y; interpolation is reliable, extrapolation isn't.
Calculus
- Stationary points: f'(x) = 0. Second derivative test: f'' > 0 minimum, f'' < 0 maximum.
- Points of inflection: f''(x) = 0 and f'' changes sign.
- Increasing/decreasing: f'(x) > 0 / f'(x) < 0.
- Tangent and normal: y − y₁ = m(x − x₁), normal gradient −1/m.
- Reverse chain rule: ∫f(ax + b) dx = (1/a)F(ax + b) + C, e.g. ∫e3x dx = ⅓e3x + C.
- Area between two curves: ∫ ǀf(x) − g(x)ǀ dx between the intersection points.
- Kinematics: v = ds/dt; speed = ǀvǀ; the particle changes direction where v = 0 and changes sign.
3What you must memorise: HL extras HL
- Polynomials: factor theorem (p(a) = 0 ⇔ (x − a) is a factor); remainder theorem (remainder = p(a)); complex roots of real polynomials come in conjugate pairs.
- Complex numbers: z z* = ǀzǀ², ǀz₁z₂ǀ = ǀz₁ǀǀz₂ǀ, arg(z₁z₂) = arg z₁ + arg z₂; the n-th roots of unity are spaced 2π/n apart.
- Proof: the structure of induction and of proof by contradiction.
- Vectors: angle between a line and a plane uses sin θ with the normal; perpendicular vectors have v · w = 0; parallel vectors have v × w = 0.
- Calculus: implicit differentiation, related rates (chain rule with dt), l'Hôpital's rule (only for 0/0 or ∞/∞), integration by substitution, partial fractions.
- Useful integrals not given: ∫sec²x dx = tan x + C; ∫tan x dx = −ln ǀcos xǀ + C (or ln ǀsec xǀ + C).
- Continuous random variables: the pdf integrates to 1; the median m satisfies ∫ f(x) dx = 0.5 up to m; the mode is where f(x) is greatest.
- Differential equations: separating variables; the substitution y = vx for homogeneous equations.
- Maclaurin series by substitution: ex² = 1 + x² + x⁴/2! + …, using the booklet's eˣ series.
4Using the booklet fast
- Learn the layout during the course. The topics follow the syllabus order (1 to 5), and each formula is labelled with its syllabus section number.
- Write the formula in the context of the question before substituting. Examiners give method marks for, e.g., "a/sin 40° = 12/sin 75°", not for the general formula copied out.
- Check the conditions printed next to formulas: r ≠ 1, ǀrǀ < 1, θ in radians, f(x) > 0 for area.
Common mistakeSpending exam time searching for a formula that isn't in the booklet. Learn the lists above so you know straight away.
Linking questionFor each topic in full see Trig identities, equations and graphs, Differentiation and Random variables, binomial and normal distributions. The AI version is the AI formula booklet guide.
Test yourself — cover the answers first
Quick check
- Is the second derivative test in the AA formula booklet?
- Is the period of tan(bx) 2π/b?
- Which trig identities does the AA SL booklet give?
- Is Bayes' theorem in the SL booklet?
- HL Is ∫sec²x dx given in the HL booklet?
Show answers
- no
- no, it is π/b
- tan θ = sin θ/cos θ, cos²θ + sin²θ = 1, sin 2θ and cos 2θ
- no, only in the HL booklet
- no, but d/dx tan x = sec²x is, so it follows
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