A Level Further Mathematics CCEA
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118 topics in 43 modules
☑️ Matrices (Pure Mathematics) 8 topics
- Matrices
- Matrix Transformations
- Determinants
- Inverting Matrices
- Matrices and Simultaneous Equations
- Factorising Determinants
- Eigenvectors and Eigenvalues
- Diagonalisation
☑️ Vectors (Pure Mathematics) 5 topics
- Use of Vectors in i, j and k unit
- Scalar Products
- Plane Geometry
- Intersections and Distances
- Vector Product
☑️ Hooke's Law (Applied Mathematics) 2 topics
- Modulus of Elasticity
- Applicatoin of Hooke's Law
☑️ Work and Energy (Applied Mathematics) 4 topics
- Elastic Energy
- Kinetic Enrgy
- Gravitatinal Potential Energy
- Principle of Conservation of Mechanical Energy
☑️ Power (Applied Mathematics) 2 topics
- Definition of Power
- Solve Problems using Power
☑️ Circular Motion (Applied Mathematics) 2 topics
- Angular Speed
- Acceleration of a Particle moving in a Circle
☑️ Further Particle Equilibrum (Applied Mathematics) 1 topic
- Solve more complex problems involving particle equilibrium
☑️ Resultant and Relative Velocity (Applied Mathematics) 2 topics
- Solve problems involving resultant velocity
- Solve problems involving relative velocity
☑️ Gravitation (Applied Mathematics) 2 topics
- Use of the universal law of gravitation
- Solve problems involving satellite motion
☑️ Dimensions (Applied Mathematics) 1 topic
- Dimensional Consistency
☑️ Sampling (Applied Mathematics) 1 topic
- Understanding of sampling methods
☑️ Probability (Applied Mathematics) 2 topics
- Use of Probability formulas
- Evaluate Probability
☑️ Statistical Distributions (Applied Mathematics) 4 topics
- Geometric Distribution
- Discree Probability Model
- Calculating Probabilities with a continuous random variable
- Poisson Distributions
☑️ Bivariate Distributions (Applied Mathematics) 4 topics
- Product moment correlation coefficient
- Regression lines
- Independent and Dependent Variables
- Dangers of Extrapolation
☑️ Group Theory (Applied Mathematics) 3 topics
- Binary Operations
- Lagrange's Theorem
- Isomorphism
☑️ Algorithms on Graphs (Applied Mathematics) 4 topics
- Solve Problems involving Critical Path Analysis
- Prim's Algorithm
- Use of Binary Trees
- Dijkstra's Algorithm
☑️ Recurrence Relationships (Applied Mathematics) 2 topics
- Recurrence Relationships
- Solve homogenous, constant coefficient and linear recurrence relations
☑️ Boolean Algebra (Applied Mathematics) 1 topic
- Use of Truth Tables
☑️ Proof (Pure Mathematics) 1 topic
- Construct Proof using mathematical induction
☑️ Further algebra and functions (Pure Mathematics) 4 topics
- Decompose rational functions into partial fractions
- Relationship between roots and coefficents of quadratic equations
- Use of formulae for the sums of integers, squares and cubes
- Mclaurin series of a function
☑️ Complex numbers (Pure Mathematics) 9 topics
- De Moivre's Theorem
- Solving Quadratic Equations
- nth theory
- Solving cubic or quartic equations with real coefficients
- Fucntions with complex numbers in the form x + iy with x and y real;
- Use and Interpret Agrand diagrams
- Radian and Measure for angles
- Use the modulus-argument form of a complex number
- Use and Conversion of the Cartesian form
☑️ Further Calculus (Pure Mathematics) 3 topics
- Calculus with Inverse Trig Functions
- Integration with Partial Fractions
- Reduction Formulas
☑️ Polar co-ordinates (Pure Mathematics) 5 topics
- Polar co-ordinates
- Use of Polar Coordinates
- Conversion between Polar and Cartesian coordinates
- Sketching curves
- Area
☑️ Hyperbolic functions (Pure Mathematics) 3 topics
- Understanding of definitions of hyperbolic functions
- Differentiate and integrate hyprbolic functions
- Inverse hyperbolic functions
☑️ Differential equations (Pure Mathematics) 2 topics
- General and Particular solutions to differential equations
- Use differential equations in modelling
☑️ Simple Harmonic Motion (Applied Mathematics) 3 topics
- Solve equations using Simple Harmonic motions
- Solve problems involving the simple pendulum
- Solve problems involving osciliation of a particle attached to the end of an elastic spring or string
☑️ Damped Oscilliations (Applied Mathematics) 1 topic
- odel damped oscilliations using second order differential equations
☑️ Centre of Mass (Applied Mathematics) 3 topics
- Find the centre of mass of systems
- Find the centre of mass of rectangular, triangular and circular laminae
- Find the centre of mass of composite laminae
☑️ Frameworks (Applied Mathematics) 1 topic
- Solve problems using Frameworks
☑️ Further Circular Motion (Applied Mathematics) 2 topics
- Solve more complex problems onvolving motion
- Solve problems involving motion in a vertical cirlce
☑️ Further Kinematics (Applied Mathematics) 2 topics
- Solve problems using i, j and k unit vectors
- Solve problems involving variable acceleration
☑️ Further Centre of Mass (Applied Mathematics) 4 topics
- Find the centre of mass of laminae and solids
- Find the centre of mass of composite bodies
- Solve problems involving suspended bodies
- Solve sliding/toppling problems
☑️ Force systems in two dimensions (Applied Mathematics) 1 topic
- Find the general resultant of a system of coplanar forces
☑️ Restitution (Applied Mathematics) 1 topic
- Use of Newton's Law of Restitution
☑️ Linear combinations of independent variables (Applied Mathematics) 1 topic
- Linear combinations of independent variables
☑️ Sampling and estimation (Applied Mathematics) 3 topics
- Central Limit Theorem
- Standard error of the mean
- Calculate confidence intervals
☑️ The t-distribution (Applied Mathematics) 2 topics
- Hypothesis tests
- Formulatoin of a hypothesis
☑️ 𝝌𝟐 tests (Applied Mathematics) 2 topics
- Fitting a theoretical distribution to given data
- Use of an 𝝌𝟐 test
☑️ Counting (Applied Mathematics) 2 topics
- Principle of Inclusion and Exclusion
- Enumeration of restricted positions, including the use of Rook polynominals
☑️ Graph Theory (Applied Mathematics) 4 topics
- Bipartite Graphs
- Understanding of basic graphs and language
- Hall's marriage theorem
- Understanding of trees
☑️ Algorithms on graphs (Applied Mathematics) 3 topics
- Hamiltonian Cycle
- PERT
- Solve variable linear programming problems
☑️ Generating functions (Applied Mathematics) 2 topics
- Solve simple summation problems
- Proof of simple formulae
☑️ Group theory (Applied Mathematics) 4 topics
- Working with Symmetry groups
- Understand the concept of a cycle index
- Use a table of cycle indices
- Use Polya's Enumeration Theorem
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A Level Further Mathematics CCEA Revision Content
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A Level Further Mathematics CCEA - Matrices (Pure Mathematics) - Matrices Content Preview
Matrices (Pure Mathematics)
Matrices
Introduction to Matrices
- A matrix is a rectangular arrangement of numbers.
- These numbers are called the elements or entries of the matrix.
- Rows run horizontally and columns run vertically in a matrix.
- The dimension of a matrix is given in the form 'm x n' (m rows by n columns).
- A square matrix has the same number of rows as it does columns.
- An identity matrix is a square matrix with ones along the main diagonal and zeroes elsewhere.
Matrix Operations
- Matrices can be added or subtracted element by element if they are of the same dimensions.
- Matrix multiplication is possible if the number of columns in the first matrix equals the number of rows in the second.
- In the multiplication of matrices, order matters (AB ≠ BA).
- The transpose of a matrix is obtained by swapping the rows and columns.
- A matrix can be multiplied by a scalar; each element is multiplied by the scalar.
Determinants and Inverses
- The determinant of a matrix is a specific number that can only be calculated for square matrices.
- If the determinant of a matrix is zero, it is said to be singular or non-invertible.
- The inverse of a matrix A is denoted as A^-1, and it satisfies the property that AA^-1 = A^-1A = I, where I is the identity matrix.
- To find the inverse of a matrix, one has to use the formula A^-1 = 1/det(A) adj(A), where adj(A) is the adjugate of A.
Applications of Matrices
- Matrices are used to solve systems of linear equations through Gaussian elimination or Cramer's rule.
- They are used to perform linear transformations including rotation, dilation, reflection and shearing.
- In computer graphics, matrices are used to manipulate and transform 3D figures.
- In probability and statistics, matrices are used in Markov chains to model and predict behaviour over time.
Question: State the condition on the determinant of a square matrix \(A\) for \(A^{-1}\) to exist.
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