Syllabus

IB Mathematics AA HL/SL Syllabus Overview

This is based on the IB Mathematics: Analysis and Approaches syllabus (first assessment 2021).

Intuition

The IB Mathematics AA syllabus is structured like a journey from concrete to abstract. Number and Algebra provides the computational foundation. Functions introduce the idea of transformation and mapping. Trigonometry extends algebra into circular and periodic phenomena. Calculus reveals how to measure change and accumulation. Statistics and Probability quantify uncertainty. Vectors and Matrices describe spatial relationships and linear transformations. The syllabus mirrors how mathematical thought evolved historically, from solving practical problems to abstract reasoning.

Topic	Subtopic	Level	Key Understanding	Relevant Equations & Identities
1. Number and Algebra	Arithmetic Sequences	SL & HL	Sequences with a constant common difference; finding the nth term and the sum of the first n terms.	`u_n = u_1 + (n-1)d`
	Arithmetic Series	SL & HL	Sum formulae for arithmetic series in terms of first term, last term, common difference, and number of terms.	`S_n = (n/2)(2u_1 + (n-1)d) = (n/2)(u_1 + u_n)`
	Geometric Sequences	SL & HL	Sequences with a constant common ratio; finding the nth term.	`u_n = u_1 r^{n-1}`
	Geometric Series	SL & HL	Finite and infinite geometric series; convergence conditions for infinite series.	`S_n = u_1(1 - r^n)/(1 - r)`; `S_inf = u_1/(1 - r)` for `
	Sigma Notation	SL & HL	Using summation notation to express and evaluate series; rules for manipulating sums.	`\sum_{r=1}^n u_r`; `\sum_{r=1}^n k = kn`; `\sum_{r=1}^n (a_r + b_r) = \sum a_r + \sum b_r`
	The Binomial Theorem	SL & HL	Expansion of `(a + b)^n` using binomial coefficients; Pascal’s triangle and the relationship to combinations.	`(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r`; `\binom{n}{r} = n! / (r!(n-r)!)`
	Applications of the Binomial Theorem	SL & HL	Finding specific terms in a binomial expansion without fully expanding.	General term: `T_{r+1} = \binom{n}{r} a^{n-r} b^r`
	Compound Interest	SL & HL	Calculating the future value of an investment or loan with compounding.	`FV = PV(1 + r)^n` where `r` is the rate per period and `n` is the number of periods
	Compound Depreciation	SL & HL	Calculating the depreciated value of an asset over time.	`FV = PV(1 - r)^n`
	Permutations	HL Only	Counting ordered arrangements of objects; counting with and without repetition.	`^nP_r = n!/(n-r)!`
	Combinations	HL Only	Counting unordered selections; applications to probability and combinatorial problems.	`^nC_r = n!/(r!(n-r)!)`
	Complex Numbers: Introduction	HL Only	The imaginary unit `i` where `i^2 = -1`; complex numbers in Cartesian form `a + bi`.	`z = a + bi`; `i^2 = -1`; `i^3 = -i`; `i^4 = 1`
	Complex Numbers: Operations	HL Only	Addition, subtraction, multiplication, and division of complex numbers; equating real and imaginary parts.	`(a+bi)(c+di) = (ac-bd) + (ad+bc)i`; division by multiplying by the conjugate
	Modulus of a Complex Number	HL Only	The modulus is the distance from the origin to the point `(a, b)` on the Argand diagram.	`
	Argument of a Complex Number	HL Only	The argument is the angle from the positive real axis to the line joining the origin to the point `(a, b)`.	`\arg(z) = \arctan(b/a)` with quadrant adjustment; principal argument `-\pi < \theta \le \pi`
	Polar Form	HL Only	Representing complex numbers in modulus-argument (polar) form on the Argand diagram.	`z = r(\cos\theta + i\sin\theta) = r\,\mathrm{cis}\,\theta`
	Euler Form	HL Only	Euler’s formula and the exponential representation of complex numbers.	`e^{i\theta} = \cos\theta + i\sin\theta`; `z = re^{i\theta}`; `e^{i\pi} + 1 = 0`
	De Moivre’s Theorem	HL Only	Raising complex numbers to integer powers using polar form.	`[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)`
	Roots of Complex Numbers	HL Only	Finding the nth roots of a complex number; roots lie equally spaced on a circle of radius `r^{1/n}`.	`z_k = r^{1/n}[\cos((\theta + 2k\pi)/n) + i\sin((\theta + 2k\pi)/n)]` for `k = 0, 1, \ldots, n-1`
	Complex Conjugate Root Theorem	HL Only	If a polynomial with real coefficients has a complex root `a + bi`Then `a - bi` is also a root.	If `f(a+bi) = 0` then `f(a-bi) = 0` for real coefficients
	Sum and Product of Roots	HL Only	Relationship between coefficients and sums/products of roots for polynomial equations.	Quadratic: sum = `-b/a`Product = `c/a`; Cubic: sum = `-b/a`Sum of pairwise = `c/a`Product = `-d/a`
	Proof by Contradiction	HL Only	Assuming the negation of a statement and arriving at a contradiction to prove the original statement.	N/A
	Proof by Mathematical Induction	HL Only	Proving statements for all positive integers using the induction principle: base case, inductive hypothesis, inductive step.	N/A
	Counterexamples	HL Only	Using a single counterexample to disprove a universal statement.	N/A
	Direct Variation	HL Only	Modelling directly proportional relationships between two variables.	`y = kx`; `y_1/x_1 = y_2/x_2`
	Inverse Variation	HL Only	Modelling inversely proportional relationships between two variables.	`y = k/x`; `x_1 y_1 = x_2 y_2`
2. Functions	Linear Functions	SL & HL	Equations and graphs of straight lines; gradient, intercepts, parallel and perpendicular lines.	`y = mx + c`; `m = (y_2 - y_1)/(x_2 - x_1)`; perpendicular gradients: `m_1 m_2 = -1`
	Quadratic Functions: Standard Form	SL & HL	`f(x) = ax^2 + bx + c`; identifying the y-intercept and direction of opening.	y-intercept at `(0, c)`; opens up if `a > 0`Down if `a < 0`
	Quadratic Functions: Vertex Form	SL & HL	`f(x) = a(x - h)^2 + k`; identifying the vertex, axis of symmetry, and optimal value.	Vertex at `(h, k)`; axis of symmetry `x = h`
	Quadratic Functions: Discriminant	SL & HL	The discriminant determines the number and nature of roots.	`D = b^2 - 4ac`; `D > 0`: two distinct real roots; `D = 0`: one repeated root; `D < 0`: no real roots
	Exponential Functions	SL & HL	Properties and graphs; exponential growth and decay; the natural exponential function `e^x`.	`y = a^x`; `y = e^x`; horizontal asymptote at `y = 0`
	Logarithmic Functions	SL & HL	Definition as the inverse of exponentiation; domain and range; graphs of `\log_a x`.	`y = \log_a x` iff `a^y = x`; domain `x > 0`; passes through `(1, 0)`
	Laws of Logarithms	SL & HL	Product, quotient, and power rules for logarithms; change of base formula.	`\log_a(xy) = \log_a x + \log_a y`; `\log_a(x/y) = \log_a x - \log_a y`; `\log_a(x^n) = n\log_a x`
	Solving Equations	SL & HL	Solving equations analytically and graphically; intersection of graphs as solutions to `f(x) = g(x)`.	N/A
	Domain and Range	SL & HL	Identifying the domain (set of inputs) and range (set of outputs) of functions from their equations and graphs.	N/A
	Even and Odd Functions	SL & HL	Symmetry properties: even functions are symmetric about the y-axis; odd functions have rotational symmetry about the origin.	Even: `f(-x) = f(x)`; Odd: `f(-x) = -f(x)`
	Horizontal Translations	SL & HL	Shifting graphs left or right by a constant.	`y = f(x - b)` shifts right by `b` units; `y = f(x + b)` shifts left by `b` units
	Vertical Translations	SL & HL	Shifting graphs up or down by a constant.	`y = f(x) + c` shifts up by `c` units; `y = f(x) - c` shifts down by `c` units
	Reflections	SL & HL	Reflecting graphs in the x-axis, y-axis, and the line `y = x`.	`y = -f(x)` (x-axis); `y = f(-x)` (y-axis); `y = f^{-1}(x)` (line `y = x`)
	Stretches	SL & HL	Vertical and horizontal stretches and compressions of graphs.	`y = af(x)` (vertical stretch by factor `a`); `y = f(bx)` (horizontal stretch by factor `1/b`)
	Composite Functions	SL & HL	Combining two or more functions; finding the domain of a composite function.	`(f \circ g)(x) = f(g(x))`; domain of `f \circ g` is the set of `x` in domain of `g` where `g(x)` is in domain of `f`
	Inverse Functions	SL & HL	Finding the inverse function algebraically; reflecting the graph of `f` in `y = x` to obtain `f^{-1}`.	`f^{-1}(f(x)) = x`; domain of `f^{-1}` = range of `f`; range of `f^{-1}` = domain of `f`
	Polynomial Functions	SL & HL	Identifying polynomials by degree; determining end behaviour, intercepts, and turning points; sketching graphs.	`f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0`
	Linear Inequalities	SL & HL	Solving and representing solutions to linear inequalities on a number line.	`ax + b > 0`; multiply/divide by negative reverses the inequality
	Quadratic Inequalities	SL & HL	Solving quadratic inequalities using the discriminant and sign analysis.	Sign analysis of `(x - \alpha)(x - \beta) > 0` or `< 0`
	Rational Inequalities	SL & HL	Solving inequalities involving rational expressions; critical values and test intervals.	Find zeros of numerator and denominator; test intervals; exclude values making denominator zero
	Factor Theorem	HL Only	If `(x - a)` is a factor of `f(x)`Then `f(a) = 0`; used to factorise polynomials.	`f(a) = 0` implies `(x - a)` divides `f(x)`; find integer roots among factors of constant term
	Remainder Theorem	HL Only	The remainder when `f(x)` is divided by `(x - a)` equals `f(a)`.	`f(x) = (x - a)Q(x) + f(a)`; remainder when dividing by `(ax - b)` is `f(b/a)`
	Rational Functions	HL Only	Functions of the form `f(x) = p(x)/q(x)`; identifying vertical, horizontal, and oblique asymptotes.	Vertical asymptotes where `q(x) = 0`; horizontal asymptote from `p(x)/q(x)` as `x \to \pm\infty`
	Domain Restriction for Inverses	HL Only	Restricting the domain of a non-injective function so that its inverse exists and is also a function.	N/A
	Self-Inverse Functions	HL Only	Functions for which `f = f^{-1}`; their graphs are symmetric about `y = x`.	`f(f(x)) = x`; examples: f(x) = a - x``f(x) = 1/x
	Reciprocal Function Graphs	HL Only	Sketching `y = 1/f(x)` from the graph of `y = f(x)`; new asymptotes at zeros of `f(x)`.	N/A
	Absolute Value Graphs	HL Only	Sketching `y =	f(x)
	Modulus Equations	HL Only	Solving equations of the form `	f(x)
	Modulus Inequalities	HL Only	Solving inequalities involving absolute value.	`
3. Geometry and Trigonometry	Distance in 3D Space	SL & HL	Calculating the distance between two points in three-dimensional space.	`d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}`
	Midpoint in 3D Space	SL & HL	Finding the midpoint of a line segment in 3D.	`M = ((x_1+x_2)/2, (y_1+y_2)/2, (z_1+z_2)/2)`
	Volume and Surface Area	SL & HL	Volume and surface area of cuboid, cylinder, cone, sphere, hemisphere, and composite solids.	`V_{\mathrm{cyl}} = \pi r^2 h`; `V_{\mathrm{cone}} = (1/3)\pi r^2 h`; `V_{\mathrm{sphere}} = (4/3)\pi r^3`; `SA_{\mathrm{sphere}} = 4\pi r^2`
	Right-Angled Trigonometry	SL & HL	SOH-CAH-TOA; finding unknown sides and angles in right-angled triangles.	`\sin\theta = \mathrm{opp}/\mathrm{hyp}`; `\cos\theta = \mathrm{adj}/\mathrm{hyp}`; `\tan\theta = \mathrm{opp}/\mathrm{adj}`
	Trigonometric Functions and Graphs	SL & HL	Graphs of sin, cos, tan; period, amplitude, phase shift, and vertical shift.	`y = a\sin(b(x - c)) + d`; period = `2\pi/
	Sine Rule	SL & HL	Finding unknown sides and angles in any triangle.	`a/\sin A = b/\sin B = c/\sin C`; ambiguous case for SSA
	Cosine Rule	SL & HL	Finding unknown sides and angles; generalised Pythagorean theorem for any triangle.	`a^2 = b^2 + c^2 - 2bc\cos A`; `A = \cos^{-1}((b^2 + c^2 - a^2)/(2bc))`
	Area of a Triangle	SL & HL	Formula using two sides and the included angle.	`A = (1/2)ab\sin C`
	Pythagorean Identity	SL & HL	Fundamental trigonometric identity relating sin and cos.	`\sin^2\theta + \cos^2\theta = 1`
	Double Angle Identities	SL & HL	Expressions for \sin 2\theta``\cos 2\thetaAnd `\tan 2\theta` in terms of `\sin\theta` and `\cos\theta`.	`\sin 2\theta = 2\sin\theta\cos\theta`; `\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta`
	Vectors as Column Vectors	SL & HL	Representing vectors in component form; adding, subtracting, and scalar multiplication.	`\mathbf{v} = \begin`{pmatrix}`a \\ b \end`{pmatrix}; `\mathbf{v} + \mathbf{w} = \begin`\{pmatrix}` a+c \\ b+d \end`\{pmatrix}
	Magnitude of a Vector	SL & HL	Calculating the length (magnitude) of a vector in 2D and 3D.	`
	Unit Vectors	SL & HL	Vectors with magnitude 1; expressing a vector as a scalar multiple of a unit vector.	`\hat{\mathbf{v}} = \mathbf{v}/
	Scalar (Dot) Product	SL & HL	Definition in terms of components and in terms of magnitudes and angle; geometric interpretation as projection.	`\mathbf{v} \cdot \mathbf{w} = a_1 b_1 + a_2 b_2` (+ `a_3 b_3`); `\mathbf{v} \cdot \mathbf{w} =
	Angle Between Two Vectors	SL & HL	Using the scalar product to determine the angle between two vectors.	`\cos\theta = (\mathbf{v} \cdot \mathbf{w}) / (
	Perpendicularity of Vectors	SL & HL	Two vectors are perpendicular if and only if their scalar product is zero.	`\mathbf{v} \perp \mathbf{w}` iff `\mathbf{v} \cdot \mathbf{w} = 0`
	Vector Equation of a Line	SL & HL	Lines in two and three dimensions expressed in vector form with a position vector and direction vector.	`\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}` where `\mathbf{a}` is on the line, `\mathbf{b}` is the direction vector
	Parametric Line Equations	SL & HL	Expressing the coordinates of points on a line in terms of a parameter.	x = x_0 + \lambda a``y = y_0 + \lambda b``z = z_0 + \lambda c
	Cartesian (Symmetric) Line Equations	SL & HL	Eliminating the parameter to obtain the symmetric form of a line equation.	`(x - x_0)/a = (y - y_0)/b = (z - z_0)/c`
	Cross Product (Vector Product)	HL Only	The vector product of two vectors in 3D; result is a vector perpendicular to both inputs.	`\mathbf{v} \times \mathbf{w}` computed via determinant of a 3x3 matrix
	Properties of the Cross Product	HL Only	Anti-commutativity; magnitude gives area of parallelogram; scalar triple product gives volume of parallelepiped.	`\mathbf{v} \times \mathbf{w} = -(\mathbf{w} \times \mathbf{v})`; `
	Equation of a Plane	HL Only	Cartesian, vector, and parametric forms of a plane in 3D.	`\mathbf{r} \cdot \mathbf{n} = d`; `ax + by + cz = d`; `\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}` (parametric)
	Intersection of Line and Plane	HL Only	Finding the point of intersection; determining whether a line is parallel to or lies in a plane.	Substitute parametric line equations into plane equation; solve for `\lambda`
	Angle Between Line and Plane	HL Only	Finding the angle between a line and a plane using the direction vector and the normal vector.	`\sin\theta =
	Angle Between Two Planes	HL Only	Finding the angle between two planes using their normal vectors.	`\cos\theta =
	Complex Numbers on the Argand Diagram	HL Only	Representing complex numbers geometrically; the Argand plane with real and imaginary axes.	`z = a + bi` plotted at `(a, b)`; multiplication: add arguments, multiply moduli
	Compound Angle Identities	HL Only	Formulae for sine, cosine, and tangent of the sum and difference of two angles.	`\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta`; `\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta`
	Reciprocal Trigonometric Functions	HL Only	Definitions and identities for sec, cosec, and cot.	`\sec\theta = 1/\cos\theta`; `\cosec\theta = 1/\sin\theta`; `\cot\theta = \cos\theta/\sin\theta`
	Matrix Addition and Scalar Multiplication	HL Only	Adding and subtracting matrices of the same order; multiplying a matrix by a scalar.	`(A + B)_`{ij}`= A_`{ij}`+ B_`{ij}; `(kA)_`\{ij}` = k \cdot A_`\{ij}
	Matrix Multiplication	HL Only	Multiplying matrices; conditions for multiplication; identity matrix.	`(AB)_`{ij}`= \sum_k A_`{ik}`B_`{kj}“; `AI = IA = A`; matrix multiplication is not commutative
	Determinants (2x2 and 3x3)	HL Only	Calculating determinants; geometric interpretation as area/volume scale factor.	`\det\begin`{pmatrix}`a & b \\ c & d \end`{pmatrix}`= ad - bc`; 3x3 by cofactor expansion
	Inverse of a 2x2 Matrix	HL Only	Formula for the inverse of a 2x2 matrix; verifying by multiplication.	`\begin`{pmatrix}`a & b \\ c & d \end`{pmatrix}`^{-1} = \frac{1}{ad-bc}\begin`{pmatrix}`d & -b \\ -c & a \end`{pmatrix}“
	Inverse of a 3x3 Matrix	HL Only	Finding the inverse using the adjugate (adjoint) matrix method.	`A^{-1} = (1/\det A) \cdot \mathrm{adj}(A)`; `A^{-1}` exists iff `\det A \ne 0`
	Solving Systems with Inverse Matrices	HL Only	Solving `A\mathbf{x} = \mathbf{b}` by pre-multiplying by `A^{-1}`.	`\mathbf{x} = A^{-1}\mathbf{b}`; requires `\det A \ne 0` (unique solution)
	Augmented Matrices and Row Reduction	HL Only	Solving systems by Gaussian elimination; row operations on augmented matrices.	Row operations: swap rows, multiply row by scalar, add multiple of one row to another
	Eigenvalues	HL Only	Scalar values `\lambda` satisfying `A\mathbf{v} = \lambda\mathbf{v}`; found by solving the characteristic equation.	`\det(A - \lambda I) = 0`; for 2x2: `\lambda^2 - \mathrm{tr}(A)\lambda + \det(A) = 0`
	Eigenvectors	HL Only	Non-zero vectors `\mathbf{v}` satisfying `A\mathbf{v} = \lambda\mathbf{v}`; finding for each eigenvalue.	Solve `(A - \lambda I)\mathbf{v} = \mathbf{0}` for each `\lambda`; eigenvectors are unique up to scalar multiples
4. Statistics and Probability	Populations and Samples	SL & HL	Distinguishing between populations (entire group) and samples (subset); representative sampling.	N/A
	Sampling Methods	SL & HL	Simple random, systematic, stratified, and quota sampling; convenience, judgement, and voluntary response.	N/A
	Reliability and Bias	SL & HL	Identifying sources of bias; understanding the effect of sample size on reliability.	N/A
	Measures of Central Tendency	SL & HL	Mean, median, and mode; their relative merits for different data types (discrete vs continuous, skewed data).	`\bar{x} = \sum x_i / n`; median = middle value; mode = most frequent value
	Grouped Data	SL & HL	Estimating the mean from grouped data using mid-interval values.	`\bar{x} \approx \sum(f_i x_i) / \sum f_i` where `x_i` is the midpoint of class `i`
	Variance and Standard Deviation	SL & HL	Measuring the spread of data around the mean; using the unbiased estimator for sample variance.	`s^2 = \sum(x_i - \bar{x})^2 / (n-1)`; `s = \sqrt{s^2}`; `IQR = Q_3 - Q_1`
	Histograms	SL & HL	Frequency histograms with equal and unequal class widths; frequency density.	Frequency density = frequency / class width; area of bar = frequency
	Box-and-Whisker Plots	SL & HL	Five-number summary (min, Q1, median, Q3, max); identifying outliers and comparing distributions.	Outlier: value `< Q_1 - 1.5 \times IQR` or `> Q_3 + 1.5 \times IQR`
	Cumulative Frequency Graphs	SL & HL	Constructing ogives; estimating median, quartiles, and percentiles from the graph.	N/A
	Set Theory and Notation	SL & HL	Universal set, subsets, union, intersection, complement; two-set and three-set Venn diagrams.	`P(A \cup B) = P(A) + P(B) - P(A \cap B)`; `P(A') = 1 - P(A)`
	Conditional Probability	SL & HL	The probability of an event given that another event has occurred; restricted sample space.	`P(A
	Independence of Events	SL & HL	Two events are independent if the occurrence of one does not affect the probability of the other.	`A \perp B` iff `P(A \cap B) = P(A)P(B)` iff `P(A
	Tree Diagrams and Sample Spaces	SL & HL	Using tree diagrams for sequential events; exhaustive sample spaces for equally likely outcomes.	N/A
	Discrete Random Variables	SL & HL	Random variables that take countable values; probability mass functions (PMF).	Conditions: `0 \le p(x) \le 1` for all `x`; `\sum p(x) = 1`
	Expected Value	SL & HL	The long-run average value of a discrete random variable.	`E(X) = \sum x_i p(x_i)`; linearity: `E(aX + b) = aE(X) + b`
	Variance of a Discrete Random Variable	SL & HL	Measuring the spread of a discrete random variable about its mean.	`Var(X) = E(X^2) - [E(X)]^2`; `Var(aX + b) = a^2 Var(X)`
	Binomial Distribution	SL & HL	Modelling the number of successes in `n` independent Bernoulli trials with probability `p`.	`X \sim B(n, p)`; `P(X = x) = \binom{n}{x}p^x(1-p)^{n-x}`; `E(X) = np`; `Var(X) = np(1-p)`
	Normal Distribution	SL & HL	Properties of the normal curve; symmetric about `\mu`; approximately 68-95-99.7 rule; standardisation.	`X \sim N(\mu, \sigma^2)`; `Z = (X - \mu)/\sigma` where `Z \sim N(0, 1)`
	Standard Normal Distribution	SL & HL	The standard normal `Z`; using tables or technology to find probabilities.	`P(Z < z)` from tables; `P(a < Z < b) = P(Z < b) - P(Z < a)`
	Linear Regression: Equation of Line	SL & HL	Equation of the least squares regression line of `y` on `x`; interpreting slope and y-intercept.	`y - \bar{y} = r(s_y/s_x)(x - \bar{x})`; `y = a + bx` where `b = r(s_y/s_x)`
	Pearson’s Correlation Coefficient	SL & HL	Measuring the strength and direction of linear association; interpreting `r` and `r^2`.	`r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i-\bar{x})^2 \sum(y_i-\bar{y})^2}}`; `-1 \le r \le 1`
	Bayes’ Theorem	HL Only	Reversing conditional probabilities; partitioning the sample space using the law of total probability.	`P(A
	Poisson Distribution	HL Only	Modelling the number of rare events in a fixed interval; Poisson as a limit of the binomial.	`X \sim Po(\lambda)`; `P(X = x) = e^{-\lambda}\lambda^x / x!`; `E(X) = Var(X) = \lambda`
	Probability Density Functions	HL Only	Continuous analogue of a PMF; `f(x) \ge 0` and total area under the curve equals 1.	`P(a \le X \le b) = \int_a^b f(x)\,dx`; `\int_{-\infty}^{\infty} f(x)\,dx = 1`
	Cumulative Distribution Functions	HL Only	`F(x) = P(X \le x)`; the CDF is the integral of the PDF; `F'(x) = f(x)`.	`F(x) = \int_{-\infty}^x f(t)\,dt`; `P(a < X < b) = F(b) - F(a)`
	Expected Value (Continuous)	HL Only	The mean of a continuous random variable; calculated by integration.	`E(X) = \int_{-\infty}^{\infty} xf(x)\,dx`; `E(X^2) = \int_{-\infty}^{\infty} x^2 f(x)\,dx`
	Variance (Continuous)	HL Only	The variance of a continuous random variable.	`Var(X) = E(X^2) - [E(X)]^2`
	Median, Mode, Quartiles (Continuous)	HL Only	Finding the median, mode, and quartiles for continuous distributions.	Median: solve `F(m) = 0.5`; `Q_1`: solve `F(q) = 0.25`; `Q_3`: solve `F(q) = 0.75`
	Linear Transformation of a Variable	HL Only	Effect of `Y = aX + b` on mean, variance, and standard deviation.	`E(Y) = aE(X) + b`; `Var(Y) = a^2 Var(X)`; `\sigma_Y =
	Linear Combinations of Independent Variables	HL Only	Mean and variance of sums and differences of independent random variables.	`E(aX + bY) = aE(X) + bE(Y)`; `Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)` for independent `X, Y`
	Inverse Normal Distribution	HL Only	Given a probability, finding the corresponding value of `X`.	`X = \mu + Z\sigma` where `Z` is found from the inverse normal function
	Student’s t-Distribution	HL Only	Used when population standard deviation is unknown and sample size is small; degrees of freedom = `n - 1`.	CI for `\mu`: `\bar{x} \pm t_{n-1,\alpha/2} \cdot s/\sqrt{n}`
	Hypothesis Testing: Concepts	HL Only	Null and alternative hypotheses; significance level `\alpha`; p-values; Type I and Type II errors.	`H_0` vs `H_1`; reject `H_0` if p-value `< \alpha`
	Chi-Squared Test for Independence	HL Only	Testing whether two categorical variables are independent; observed vs expected frequencies.	`\chi^2 = \sum (O_i - E_i)^2 / E_i`; `E_i = (\mathrm{row total} \times \mathrm{col total}) / \mathrm{grand total}`
	Chi-Squared Goodness of Fit	HL Only	Testing whether observed data fits a specified theoretical distribution.	Same formula; df = `(k - 1 - p)` where `k` = categories, `p` = estimated parameters
5. Calculus	Informal Understanding of Limits	SL & HL	The concept of a limit; understanding behaviour as `x` approaches a value or infinity; one-sided limits.	`\lim_{x \to a} f(x) = L`; `\lim_{x \to a^+}` vs `\lim_{x \to a^-}`
	Continuity	SL & HL	A function is continuous at a point if the limit equals the function value; identifying discontinuities.	`f` is continuous at `c` if `\lim_{x \to c} f(x) = f(c)`; types: removable, jump, infinite
	Differentiation: Power Rule	SL & HL	The derivative of `x^n`; differentiating sums, differences, and constant multiples.	`f(x) = x^n \implies f'(x) = nx^{n-1}`; derivative of a constant is 0
	Derivatives of Exponential Functions	SL & HL	Differentiating `e^x` and `a^x`.	`d/dx(e^x) = e^x`; `d/dx(a^x) = a^x \ln a`; `d/dx(e^{f(x)}) = f'(x)e^{f(x)}`
	Derivatives of Logarithmic Functions	SL & HL	Differentiating `\ln x` and `\log_a x`.	`d/dx(\ln x) = 1/x`; `d/dx(\ln f(x)) = f'(x)/f(x)`
	Derivative of Trigonometric Functions	SL & HL	Differentiating \sin x``\cos xAnd `\tan x` (angles in radians).	`d/dx(\sin x) = \cos x`; `d/dx(\cos x) = -\sin x`; `d/dx(\tan x) = \sec^2 x`
	Product Rule	SL & HL	Differentiating the product of two functions.	`(uv)' = u'v + uv'`
	Quotient Rule	SL & HL	Differentiating the quotient of two functions.	`(u/v)' = (u'v - uv') / v^2`
	Chain Rule	SL & HL	Differentiating composite functions.	`dy/dx = (dy/du)(du/dx)`; `d/dx[f(g(x))] = f'(g(x)) \cdot g'(x)`
	First Derivative Test	SL & HL	Using the sign of `f'(x)` to classify stationary points as local maxima or minima.	`f'(x)` changes `+` to `-`: local max; `-` to `+`: local min; no change: point of inflection
	Second Derivative	SL & HL	Finding `f''(x)`; using it to determine concavity and points of inflection.	`f''(x) = d/dx[f'(x)]`; `f''(x) > 0`: concave up; `f''(x) < 0`: concave down
	Equation of a Tangent	SL & HL	Finding the gradient at a point and writing the equation of the tangent line.	`y - y_0 = f'(x_0)(x - x_0)`; gradient = `f'(x_0)`
	Equation of a Normal	SL & HL	The normal is perpendicular to the tangent; gradient is the negative reciprocal.	`y - y_0 = -(1/f'(x_0))(x - x_0)`; condition: `f'(x_0) \ne 0`
	Points of Inflection	SL & HL	Points where the concavity changes; second derivative changes sign.	`f''(x_0) = 0` and `f''(x)` changes sign at `x_0`
	Optimisation	SL & HL	Formulating and solving real-world problems involving maxima and minima.	Find `f'(x) = 0`Classify critical points, check endpoints if domain is restricted
	Indefinite Integration	SL & HL	Antidifferentiation; the constant of integration `C`.	`\int x^n\,dx = x^{n+1}/(n+1) + C` for `n \ne -1`; `\int 1/x,dx = \ln
	Definite Integration	SL & HL	Evaluating the definite integral using the Fundamental Theorem of Calculus.	`\int_a^b f(x)\,dx = [F(x)]_a^b = F(b) - F(a)`
	Area Under a Curve	SL & HL	Using definite integration to find the area between a curve and the x-axis.	`A = \int_a^b f(x)\,dx` when `f(x) \ge 0`; split at roots where `f(x)` changes sign
	Area Between Two Curves	SL & HL	Finding the enclosed area between two functions over a given interval.	`A = \int_a^b [f(x) - g(x)]\,dx` where `f(x) \ge g(x)` on `[a, b]`
	Trapezoidal Rule	SL & HL	Numerical approximation of a definite integral; accuracy increases with more trapezia.	`\int_a^b f(x)\,dx \approx (h/2)[y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n]` where `h = (b-a)/n`
	Kinematics: Displacement and Velocity	SL & HL	Displacement, velocity, and acceleration as functions of time.	`v = ds/dt`; `a = dv/dt = d^2s/dt^2`; `s = \int v\,dt`; `v = \int a\,dt`
	Kinematics: Total Distance Travelled	SL & HL	Finding total distance travelled from a velocity-time graph.	Total distance = `\int_a^b
	Implicit Differentiation	HL Only	Differentiating equations where `y` is defined implicitly as a function of `x`.	`d/dx[y^n] = ny^{n-1}\,dy/dx`; `d/dx[xy] = y + x\,dy/dx`
	Related Rates of Change	HL Only	Connecting the rates of change of related quantities using the chain rule.	`dy/dt = (dy/dx)(dx/dt)`; differentiate both sides of a relation with respect to `t`
	Integration by Substitution	HL Only	Using a substitution to simplify integrals; changing limits for definite integrals.	`\int f(g(x))g'(x)\,dx = \int f(u)\,du` where u = g(x)``du = g'(x)\,dx
	Integration by Parts	HL Only	Reversing the product rule; choosing `u` and `dv` strategically using LIATE.	`\int u\,dv = uv - \int v\,du`; repeated application when necessary
	Definite Integrals Using Technology	HL Only	Evaluating definite integrals numerically using a graphing calculator (GDC).	N/A
	Volumes of Revolution about the x-Axis	HL Only	Volume generated by rotating a region about the x-axis.	`V_x = \pi\int_a^b [f(x)]^2\,dx`
	Volumes of Revolution about the y-Axis	HL Only	Volume generated by rotating a region about the y-axis.	`V_y = \pi\int_c^d [g(y)]^2\,dy`
	Separable Differential Equations	HL Only	First-order ODEs of the form `dy/dx = f(x)g(y)`; separating variables and integrating.	`\int dy/g(y) = \int f(x)\,dx + C`; verify solution by substitution
	Maclaurin Series: Concept	HL Only	Expressing `f(x)` as an infinite polynomial series about `x = 0`; interval of convergence.	`f(x) = \sum_{n=0}^{\infty} f^{(n)}(0)\,x^n / n!`
	Common Maclaurin Series	HL Only	Standard Maclaurin series for e^x``\sin x``\cos x``\ln(1+x)``(1+x)^n.	`e^x = 1 + x + x^2/2! + x^3/3! + \cdots`; `\sin x = x - x^3/3! + x^5/5! - \cdots`
	Integration by Partial Fractions	HL Only	Decomposing proper rational functions into partial fractions before integrating.	`P(x)/Q(x) = A/(x-a) + B/(x-b) + \cdots`; each term integrates to a log or power
	L’Hopital’s Rule	HL Only	Evaluating limits of indeterminate forms by differentiating numerator and denominator.	`\lim_{x \to a} f(x)/g(x) = \lim_{x \to a} f'(x)/g'(x)` for `0/0` or `\infty/\infty` forms
	Improper Integrals: Infinite Limits	HL Only	Integrals where one or both limits are infinite; determining convergence or divergence.	`\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx`; converges if the limit is finite
	Improper Integrals: Unbounded Integrands	HL Only	Integrals where the integrand has a vertical asymptote within the interval.	`\int_0^1 (1/\sqrt{x})\,dx = \lim_{a \to 0^+} \int_a^1 (1/\sqrt{x})\,dx = 2`; split at the discontinuity
	Convergence of Improper Integrals	HL Only	Determining whether an improper integral converges to a finite value or diverges.	`\int_1^{\infty} (1/x^p)\,dx` converges for `p > 1`Diverges for `p \le 1`

Worked Examples

Worked Example: Arithmetic and Geometric Series (Topic 1)

An arithmetic sequence has first term 5 and common difference 3. A geometric sequence has first term 2 and common ratio 2. Find the smallest value of $n$ such that the sum of the first $n$ terms of the geometric sequence exceeds the sum of the first $n$ terms of the arithmetic sequence.

Solution

Arithmetic sum: $S_n = \dfrac{n}{2}(2(5) + (n-1)(3)) = \dfrac{n}{2}(10 + 3n - 3) = \dfrac{n(3n + 7)}{2}$
Geometric sum: $T_n = \dfrac{2(2^n - 1)}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2$
We need $2^{n+1} - 2 \gt \dfrac{n(3n + 7)}{2}$ .
Testing values:
$n = 4$ : geometric $= 30$ Arithmetic $= \dfrac{4(19)}{2} = 38$ . Geometric is smaller.
$n = 5$ : geometric $= 62$ Arithmetic $= \dfrac{5(22)}{2} = 55$ . Geometric exceeds arithmetic.
Answer: $n = 5$

Worked Example: Binomial Expansion (Topic 1)

Expand $(1 + 2x)^5$ in ascending powers of $x$ up to and including the term in $x^3$ .

Solution

Using the binomial theorem with $a = 1$$b = 2x$$n = 5$ :
$(1 + 2x)^5 = 1 + 5(2x) + 10(2x)^2 + 10(2x)^3 + \cdots$
$= 1 + 10x + 40x^2 + 80x^3 + \cdots$

Worked Example: Transformations of Functions (Topic 2)

The graph of $y = f(x)$ passes through the point $(3, 5)$ . Find the coordinates of the corresponding Point on the graph of $y = -2f(x - 1) + 3$ .

Solution

Apply transformations step by step:

$f(x) \to f(x - 1)$ : translate right by 1: $(3, 5) \to (4, 5)$
$\to 2f(x - 1)$ : vertical stretch by factor 2: $(4, 5) \to (4, 10)$
$\to -2f(x - 1)$ : reflect in $x$ -axis: $(4, 10) \to (4, -10)$
$\to -2f(x - 1) + 3$ : translate up by 3: $(4, -10) \to (4, -7)$

Answer: $(4, -7)$

Worked Example: Quadratic Inequalities (Topic 2)

Solve the inequality $x^2 - 5x + 6 \lt 0$ .

Solution

Factorise: $x^2 - 5x + 6 = (x - 2)(x - 3)$
The parabola opens upward (positive $x^2$ coefficient) with roots at $x = 2$ and $x = 3$ .
The expression is negative between the roots: $2 \lt x \lt 3$

Worked Example: Trigonometric Equations (Topic 3)

Solve $\cos 2x = \cos x$ for $0 \le x \le \pi$ .

Solution

Using $\cos 2x = 2\cos^2 x - 1$ :
$2\cos^2 x - 1 = \cos x \implies 2\cos^2 x - \cos x - 1 = 0$
Let $u = \cos x$ : $2u^2 - u - 1 = 0 \implies (2u + 1)(u - 1) = 0$
$u = -1/2$ or $u = 1$
$\cos x = -1/2 \implies x = 2\pi/3$ (only solution in $[0, \pi]$ )
$\cos x = 1 \implies x = 0$
Solutions: $x = 0, 2\pi/3$

Worked Example: Vector Line Intersection (Topic 3, HL)

Find the point of intersection of the lines $\mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$ .

Solution

Equate components:
$1 + 2\lambda = 3 + \mu$ … (i)
$2 - \lambda = \mu$ … (ii)
$3\lambda = 1 - \mu$ … (iii)
From (i) and (ii): $1 + 2\lambda = 3 + 2 - \lambda \implies 3\lambda = 4 \implies \lambda = 4/3$
From (ii): $\mu = 2 - 4/3 = 2/3$
Verify with (iii): $3(4/3) = 4$ and $1 - 2/3 = 1/3$ . These do not match, so the lines are skew (they do not intersect).

If you get this wrong, revise: Vector equations of lines and intersection conditions (Topic 3).

Worked Example: Probability with Venn Diagrams (Topic 4)

In a survey, 60 people were asked whether they liked tea ( $T$ ) and coffee ( $C$ ). 35 liked tea, 40 Liked coffee, and 15 liked both. Find the probability that a randomly chosen person liked neither.

Solution

$n(T \cup C) = n(T) + n(C) - n(T \cap C) = 35 + 40 - 15 = 60$
Everyone liked at least one, so $n((T \cup C)') = 0$ .
$P(\mathrm{neither}) = 0/60 = 0$

Worked Example: Normal Distribution (Topic 4)

Heights of a population are normally distributed with mean $170$ cm and standard deviation $8$ cm. Find The probability that a randomly selected person is between $162$ cm and $178$ cm tall.

Solution

Standardise:
$z_1 = (162 - 170)/8 = -1.00$
$z_2 = (178 - 170)/8 = 1.00$
$P(162 \lt X \lt 178) = P(-1 \lt Z \lt 1) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = 0.6826$

Worked Example: Differentiation — Quotient Rule (Topic 5)

Find the equation of the normal to the curve $y = \dfrac{x^2 + 1}{x - 1}$ at the point where $x = 2$ .

Solution

Quotient rule: $u = x^2 + 1$$v = x - 1$ . Then $u' = 2x$$v' = 1$ .
$\dfrac{dy}{dx} = \dfrac{2x(x - 1) - (x^2 + 1)}{(x - 1)^2} = \dfrac{2x^2 - 2x - x^2 - 1}{(x - 1)^2} = \dfrac{x^2 - 2x - 1}{(x - 1)^2}$
At $x = 2$ : $\dfrac{dy}{dx} = \dfrac{4 - 4 - 1}{1} = -1$
Point: $y(2) = (4 + 1)/(2 - 1) = 5$ So the point is $(2, 5)$ .
Normal gradient: $m_{\mathrm{normal}} = 1$ (negative reciprocal of $-1$ ).
Normal equation: $y - 5 = 1(x - 2) \implies y = x + 3$

Worked Example: Volume of Revolution (Topic 5, HL)

Find the volume generated when the region bounded by $y = \sqrt{x}$ The $x$ -axis, and $x = 4$ is rotated $360$ degrees about the $x$ -axis.

Solution

$V = \pi\displaystyle\int_0^4 (\sqrt{x})^2\,dx = \pi\displaystyle\int_0^4 x\,dx = \pi\left[\dfrac{x^2}{2}\right]_0^4 = \pi \times 8 = 8\pi$

Notes by Level

Standard Level (SL)

The SL course covers Topics 1-5 at a foundational level. Students are expected to understand and apply core concepts in number and algebra, functions, geometry and trigonometry, statistics and probability, and calculus. Internal assessment (IA) counts for 20% of the final grade, and the remaining 80% is assessed through two written examination papers (Paper 1: non-calculator, Paper 2: Calculator).

Higher Level (HL)

The HL course extends the SL content with additional subtopics in all five areas. HL students sit Two additional papers (Paper 3: problem-solving with extended response questions) and cover more advanced techniques including complex numbers, matrices, further calculus, and hypothesis testing. The HL course demands deeper conceptual understanding and greater fluency with algebraic Manipulation.

Links to Existing Content

Number and Algebra — arithmetic and geometric sequences, sigma notation, binomial theorem
Complex Numbers — complex numbers in Cartesian, polar, and Euler form; De Moivre’s theorem; roots of polynomials
Trigonometry — trigonometric functions, identities, sine and cosine rules
Vectors — vector operations, scalar and cross products, lines and planes
Integration — antiderivatives, definite integration, area, integration techniques
Logic (Discrete Maths) — propositional logic, truth tables, proof
Maths Index — overview of all mathematics notes

Problem Set

Number and Algebra

Find the sum of the infinite geometric series $8 + 4 + 2 + 1 + \cdots$

Solution

$u_1 = 8$$r = 4/8 = 0.5$ . Since $|r| \lt 1$ The sum converges.
$S_\infty = \dfrac{8}{1 - 0.5} = 16$

If you get this wrong, revise: Sum to infinity of geometric series (Topic 1).

Find the coefficient of $x^4$ in the expansion of $(1 - 3x)^8$ .

Solution

General term: $\dbinom{8}{r}(-3x)^r$ . For $x^4$ Need $r = 4$ .
$\dbinom{8}{4}(-3)^4 = 70 \times 81 = 5670$

If you get this wrong, revise: Binomial theorem, finding specific coefficients (Topic 1).

Functions

The function $f(x) = 2x^3 - 3x^2 - 12x + 5$ has a local maximum and a local minimum. Find the coordinates of both.

Solution

$f'(x) = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x - 2)(x + 1)$
Stationary points at $x = 2$ and $x = -1$ .
$f''(x) = 12x - 6$
At $x = -1$ : $f''(-1) = -18 \lt 0$ (local max). $f(-1) = -2 - 3 + 12 + 5 = 12$ . Point: $(-1, 12)$ .
At $x = 2$ : $f''(2) = 18 \gt 0$ (local min). $f(2) = 16 - 12 - 24 + 5 = -15$ . Point: $(2, -15)$ .

If you get this wrong, revise: First and second derivative tests (Topic 5).

Find the inverse of $f(x) = \dfrac{3x + 2}{x - 1}$ and state its domain.

Solution

Let $y = \dfrac{3x + 2}{x - 1}$ . Swap: $x = \dfrac{3y + 2}{y - 1}$ .
$x(y - 1) = 3y + 2 \implies xy - x = 3y + 2 \implies xy - 3y = x + 2$
$y(x - 3) = x + 2 \implies f^{-1}(x) = \dfrac{x + 2}{x - 3}$
Domain of $f^{-1}$ : $x \neq 3$ (i.e. $\mathbb{R} \setminus \\{3\\}$ ).

If you get this wrong, revise: Finding inverse functions and domain restrictions (Topic 2).

Geometry and Trigonometry

Triangle $ABC$ has sides $a = 9$$b = 7$$c = 5$ . Find the largest angle.

Solution

The largest angle is opposite the longest side, so angle $A$ :
$\cos A = \dfrac{b^2 + c^2 - a^2}{2bc} = \dfrac{49 + 25 - 81}{70} = \dfrac{-7}{70} = -0.1$
$A = \arccos(-0.1) = 95.7$ degrees

If you get this wrong, revise: Cosine rule for finding an angle (Topic 3).

Find the area of a triangle with sides $6$ cm and $8$ cm and an included angle of $50$ degrees.

Solution

$A = \dfrac{1}{2}ab\sin C = \dfrac{1}{2}(6)(8)\sin 50^\circ = 24 \times 0.7660 = 18.4$ cm $^2$

If you get this wrong, revise: Area of a triangle formula $A = \frac{1}{2}ab\sin C$ (Topic 3).

Probability and Statistics

A bag contains 4 red and 6 blue marbles. Three marbles are drawn without replacement. Find the probability that exactly two are red.

Solution

Total ways to choose 3 from 10: $\dbinom{10}{3} = 120$
Ways to choose 2 red from 4 and 1 blue from 6: $\dbinom{4}{2} \times \dbinom{6}{1} = 6 \times 6 = 36$
$P = 36/120 = 3/10 = 0.3$

If you get this wrong, revise: Combinations and probability (Topics 1, 4).

Scores on a test are normally distributed with $\mu = 65$ and $\sigma = 10$ . The top 10% of students receive an A grade. Find the minimum score for an A.

Solution

We need the 90th percentile: $P(X \lt c) = 0.90$
$P(Z \lt z) = 0.90 \implies z = 1.282$
$c = \mu + z\sigma = 65 + 1.282 \times 10 = 65 + 12.82 = 77.8$
Minimum score for an A: $78$ (rounding up).

If you get this wrong, revise: Inverse normal distribution and percentiles (Topic 4).

Calculus

Find $\displaystyle\int_0^{\pi/2} \sin x \cos x\,dx$ .

Solution

Let $u = \sin x$$du = \cos x\,dx$ :
$\displaystyle\int_0^{\pi/2} u\,du = \left[\dfrac{u^2}{2}\right]_0^{\pi/2} = \left[\dfrac{\sin^2 x}{2}\right]_0^{\pi/2} = \dfrac{1}{2} - 0 = \dfrac{1}{2}$

If you get this wrong, revise: Integration by substitution (Topic 5).

Find the equation of the tangent to $y = \ln(x^2 + 1)$ at $x = 1$ .

Solution

$\dfrac{dy}{dx} = \dfrac{2x}{x^2 + 1}$ (chain rule)
At $x = 1$ : gradient $= 2/2 = 1$ ; $y = \ln 2$
Tangent: $y - \ln 2 = 1(x - 1) \implies y = x - 1 + \ln 2$

If you get this wrong, revise: Chain rule and equation of a tangent (Topic 5).

Evaluate $\displaystyle\int_1^e \dfrac{\ln x}{x}\,dx$ .

Solution

Substitution: let $u = \ln x$$du = dx/x$ .
When $x = 1$$u = 0$ ; when $x = e$$u = 1$ .
$\displaystyle\int_0^1 u\,du = \left[\dfrac{u^2}{2}\right]_0^1 = \dfrac{1}{2}$

If you get this wrong, revise: Integration by substitution (Topic 5).

Solve the differential equation $\dfrac{dy}{dx} = \dfrac{x}{y}$ given that $y = 2$ when $x = 1$ .

Solution

Separate variables: $y\,dy = x\,dx$
Integrate: $\dfrac{y^2}{2} = \dfrac{x^2}{2} + C$
Apply initial condition: $4/2 = 1/2 + C \implies C = 3/2$
$y^2 = x^2 + 3$
Since $y = 2 \gt 0$ at $x = 1$ : $y = \sqrt{x^2 + 3}$

If you get this wrong, revise: Separable differential equations (HL, Topic 5).

Common Pitfalls

Misreading the question, particularly with ‘hence’ vs ‘hence or otherwise’. The former requires using previous work.
Cancelling terms instead of factors, $\frac{ab + ac}{a}$ simplifies to $b + c$ , not $bc$ .

Summary

Worked Examples

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