← Further Maths syllabus

Northern IrelandFurther Maths

AS 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.

What are complex numbers, and how do you add, multiply, divide and represent them?

The imaginary unit, arithmetic of complex numbers, the complex conjugate, the Argand diagram, modulus and argument, and complex roots of real polynomial equations occurring in conjugate pairs.
A CCEA AS Further Maths answer on the imaginary unit, adding, multiplying and dividing complex numbers, the complex conjugate, the Argand diagram, modulus and argument, and why complex roots of real polynomials occur in conjugate pairs.
12 min answer →

How do you sketch rational functions and find their stationary points and asymptotes?

Curve sketching of rational functions, finding vertical and horizontal asymptotes, oblique asymptotes, the range of values a rational function can take, and locating stationary points.
A CCEA AS Further Maths answer on sketching rational functions, finding vertical, horizontal and oblique asymptotes, determining the set of values a rational function can take, and locating stationary points to complete the sketch.
12 min answer →

How does proof by mathematical induction establish a result for all positive integers?

The principle of mathematical induction and its use to prove results about summation of series, divisibility and other statements indexed by the positive integers.
A CCEA AS Further Maths answer on the principle of mathematical induction, the base case, inductive step and conclusion, and applying induction to prove summation formulae and divisibility results for all positive integers.
12 min answer →

How do matrices encode linear transformations, and how do you invert a 2x2 matrix?

Matrix algebra including addition, multiplication and the identity, the determinant and inverse of a 2x2 matrix, and matrices as linear transformations of the plane including rotations, reflections and enlargements.
A CCEA AS Further Maths answer on matrix addition and multiplication, the identity matrix, the determinant and inverse of a 2x2 matrix, singular matrices, and how matrices represent rotations, reflections and enlargements of the plane.
12 min answer →

How are the roots of a polynomial related to its coefficients?

The relationships between the roots and coefficients of quadratic, cubic and quartic equations, and forming new equations whose roots are functions of the original roots.
A CCEA AS Further Maths answer on the relationships between roots and coefficients for quadratics, cubics and quartics, the symmetric functions of the roots, and how to form a new polynomial whose roots are functions of the original roots.
12 min answer →

How do you sum finite series using standard results and the method of differences?

Summation of finite series using the standard results for the sum of r, r squared and r cubed, and the method of differences for telescoping sums.
A CCEA AS Further Maths answer on summing finite series, the standard results for the sum of r, r squared and r cubed, manipulating sigma notation, and the method of differences for telescoping series.
12 min answer →