Further Maths lessons
Further Maths: complex numbers, matrices, further algebra, further calculus, further mechanics/statistics, and exam skills.
Key topics
- Complex numbers
- Matrices
- Further algebra
- Further calculus
- Further mechanics & statistics
- Exam skills
Lesson course
Work through each lesson, run the interactive practice, and tick it off as you go.
1. Complex numbers
Core Pure
Introduction to complex numbers
Work with i and arithmetic of complex numbers.
i = √−1. Complex numbers a+bi extend the reals and can be added, subtracted and multiplied.
Add, subtract and multiply five complex numbers.
Argand diagrams
Plot complex numbers and use modulus-argument form.
The Argand diagram plots complex numbers; the modulus is the distance from the origin and the argument is the angle.
Find the modulus and argument of three numbers.
De Moivre's theorem
Use De Moivre's theorem.
De Moivre's theorem links powers of a complex number to multiples of its argument.
Use De Moivre's theorem to find a power.
2. Matrices
Core Pure
Matrix operations
Add and multiply matrices.
Matrices can be added and multiplied; matrix multiplication is not commutative.
Multiply two pairs of matrices.
Determinants & inverses
Find determinants and inverses.
The determinant of a 2×2 matrix is ad−bc; the inverse exists only when the determinant is non-zero.
Invert two 2×2 matrices.
Transformations
Use matrices for transformations.
Matrices represent linear transformations such as rotations, reflections and enlargements.
Find the matrix for a rotation.
3. Further algebra
Core Pure
Roots of polynomials
Relate roots to coefficients.
Sums and products of roots relate to a polynomial's coefficients.
Find the sum and product of roots.
Series
Sum standard series.
Standard series can be summed using formulae for Σr, Σr² and Σr³.
Use Σr, Σr² and Σr³ formulae.
Proof by induction
Prove statements by induction.
Proof by induction uses a base case and an inductive step to prove a statement for all integers n.
Prove one summation result by induction.
4. Further calculus
Core Pure
Further differentiation
Use implicit and parametric differentiation.
Differentiate inverse trig functions and use implicit and parametric differentiation.
Differentiate one implicit relation.
Further integration
Integrate harder functions and find volumes.
Integrate using partial fractions and find volumes of revolution.
Find one volume of revolution.
Differential equations
Solve differential equations.
Solve first- and second-order differential equations that model change.
Solve one first-order differential equation.
5. Further mechanics & statistics
Applied
Momentum & collisions
Apply momentum and impulse.
Apply conservation of momentum and impulse to collisions.
Solve one collision problem.
Discrete distributions
Use discrete probability distributions.
Distributions such as the Poisson model discrete random events.
Use the Poisson distribution once.
Hypothesis testing
Test hypotheses for further distributions.
Test hypotheses using further statistical distributions and significance levels.
Run one hypothesis test.
6. Exam skills
Skills
Problem solving
Tackle multi-step problems.
Tackle multi-step problems, choosing appropriate techniques and showing full working.
Solve one unstructured problem.
Using the formula book
Apply standard results accurately.
Locate and apply standard results from the formula booklet accurately.
Find and use three formula-book results.
Mock exam
Apply skills under timed conditions.
Sit a timed paper and review it by topic afterwards.
Sit one timed paper and review it.
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