Complex Numbers

You will extend the real numbers to the complex numbers; work in algebraic, polar (trigonometric) and exponential form; draw every idea on an Argand diagram (modulus, argument, conjugate, rotation and loci); prove and use De Moivre's theorem; find nth roots and roots of unity; solve polynomial equations with complex roots; and apply complex numbers to multiple angles, trigonometric equations and alternating current.

2

Logarithmic and Exponential Functions

You will define the number e and the natural logarithm, state domains and derive the laws of logarithms; find limits and asymptotes; differentiate logarithmic and exponential functions of any base, including by logarithmic differentiation; sketch ln x and eˣ as inverse functions reflected in y = x; and model growth and decay, such as Rwanda's census population, compound interest, depreciation, earthquakes and carbon-14 dating.

3

Series

You will write sums in sigma notation and evaluate finite series, including the standard sums of r, r² and r³ and sums by the method of differences; decide whether infinite series converge, using the geometric series, the nth-term, comparison, ratio and alternating series tests; build power series, Maclaurin and Taylor series; and use series to find limits, to estimate e, π, logarithms, trigonometric values and roots, and to solve equations approximately.

4

Integration

You will reverse differentiation: use differentials; find indefinite integrals with the table of standard integrals, substitution, partial fractions, trigonometric identities and substitutions, integration by parts, reduction formulae and Maclaurin series; evaluate definite integrals as limits of sums through the fundamental theorem of calculus; and use them to find areas between curves, volumes of revolution (disk, washer and shell), arc lengths, and physical quantities such as displacement and work.

5

Differential Equations

You will form differential equations from families of curves and from real situations, classify them by order, degree and linearity, solve first order equations (separable, homogeneous and linear), solve second order linear equations with constant coefficients, homogeneous and non-homogeneous, and use them to model Newton's law of cooling and electrical circuits.

6

Intersection and Sum of Subspaces

You will test whether a subset of a vector space is a subspace, find the intersection and the sum of two subspaces with a basis for each, recognise a direct sum, and link their dimensions through Grassmann's formula dim(U + W) = dim U + dim W − dim(U ∩ W).

7

Transformation of Matrices

You will reduce a matrix to echelon form with elementary row operations, use the rank to find inverses and to decide whether a linear system has no, one or infinitely many solutions, find the kernel and range of a linear transformation, and compute eigenvalues and eigenvectors to diagonalise a matrix and raise it to any power.

8

Conics

You will see the ellipse, parabola and hyperbola as slices of a cone and as curves with a focus, a directrix and an eccentricity; identify a conic from its equation; find the elements, tangents and normals of each conic; work in polar coordinates and write the polar equations of lines, circles and conics; and use conics to model dishes, orbits and navigation.

9

Random Variables

You will describe a discrete random variable by its probability and cumulative distribution functions, compute its expected value, variance and standard deviation, use the binomial and Poisson distributions, work with the density and distribution function of a continuous random variable, and solve problems with the normal distribution.

Units