← Further Maths syllabus

Northern IrelandFurther Maths

A2 1 Pure Mathematics

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does de Moivre's theorem let you take powers and roots of complex numbers?

De Moivre's theorem, the exponential (Euler) form of a complex number, using de Moivre to derive trigonometric identities, and finding the nth roots of a complex number.
A CCEA A2 Further Maths answer on de Moivre's theorem, the exponential form of a complex number, deriving trigonometric identities such as multiple-angle formulae, and finding the nth roots of a complex number on the Argand diagram.
12 min answer →

How do you solve first-order and second-order linear differential equations?

Solving first-order linear differential equations by the integrating factor, second-order linear equations with constant coefficients via the auxiliary equation, complementary function and particular integral, and modelling with differential equations.
A CCEA A2 Further Maths answer on solving first-order linear differential equations with an integrating factor, second-order linear equations with constant coefficients using the auxiliary equation, the complementary function and particular integral, and applying them to model real situations.
13 min answer →

How do you build a Maclaurin series and evaluate improper integrals, arc lengths and surface areas?

Maclaurin series expansions of standard functions, improper integrals with infinite limits or discontinuities, the arc length of a curve, and the area of a surface of revolution.
A CCEA A2 Further Maths answer on Maclaurin series expansions of standard functions, evaluating improper integrals with infinite limits or discontinuities, and finding the arc length of a curve and the area of a surface of revolution.
12 min answer →

What are eigenvalues and eigenvectors, and how do you find them for a 2x2 matrix?

Eigenvalues and eigenvectors of a 2x2 matrix, the characteristic equation, invariant lines and lines of invariant points of a transformation, and using eigenvectors to describe the geometry of a matrix.
A CCEA A2 Further Maths answer on eigenvalues and eigenvectors of a 2x2 matrix, the characteristic equation, finding eigenvectors, invariant lines and lines of invariant points, and what they reveal about a transformation.
12 min answer →

What are the hyperbolic functions, and how do you differentiate, integrate and invert them?

The definitions of the hyperbolic functions in terms of the exponential function, their identities, derivatives and integrals, and the logarithmic forms of the inverse hyperbolic functions.
A CCEA A2 Further Maths answer on the hyperbolic functions defined from the exponential function, the identities such as cosh squared minus sinh squared equals 1, their derivatives and integrals, and the logarithmic forms of the inverse hyperbolic functions.
12 min answer →

How do polar coordinates describe curves, and how do you find the area they enclose?

Polar coordinates and their relationship to Cartesian coordinates, sketching curves given in polar form, and the area enclosed by a polar curve using the half r squared integral.
A CCEA A2 Further Maths answer on polar coordinates and their conversion to Cartesian form, sketching polar curves such as circles, cardioids and roses, and finding the area enclosed by a polar curve using the half integral of r squared.
12 min answer →