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Further Mathematics
- Further Pure Mathematics 1: This paper includes topics such as complex numbers, matrices, and further aspects of calculus.
- Further Pure Mathematics 2: This paper covers hyperbolic functions, more advanced matrices, further calculus (differentiation and integration), complex numbers, and differential equations.
- Further Mechanics: This component focuses on applied mechanics topics, including projectiles, equilibrium, circular motion, Hooke's Law, linear motion under a variable force, and momentum.
- Further Probability & Statistics: This component covers topics like probability-generating functions, as well as further mechanics and statistics topics.
Further Pure Mathematics-1
- Roots of Polynomials: Solving for roots of cubic and quadratic equations.
- Rational Functions and Graphs: Analysing the behaviour and graphing of functions with polynomial numerators and denominators.
- Matrices: Introduction to 2x2 matrices, matrix multiplication, identity, inverse, and transformations.
- Polar Coordinates: Using polar coordinates as an alternative to Cartesian coordinates.
- Vectors: Further vector operations and concepts beyond introductory levels.
- Proof by Induction: Using mathematical induction to prove statements and formulas.
Further Pure Mathematics-2
- Hyperbolic functions: Definitions, graphs, identities, and inverse hyperbolic functions.
- Matrices: Solving systems of linear equations, eigenvalues and eigenvectors, matrix powers, and geometric interpretation of matrices.
- Differentiation: Differentiating inverse and hyperbolic functions, finding second derivatives for implicit and parametric relations, and using Maclaurin series to approximate functions.
- Integration: Techniques for integrating hyperbolic functions, using reduction formulas, finding arc lengths, and calculating surface areas of revolution.
- Complex numbers: De-Moivre's theorem and its applications, multiple angle identities, and roots of unity.
- Differential equations: Solving various types of differential equations
Further Mechanics
- Motion of a projectile: This topic covers the analysis of the path of a projectile, including the equations for its trajectory and motion.
- Equilibrium of a rigid body: This involves the study of the conditions required for a rigid body to remain in static equilibrium, including the concepts of moments and the center of mass.
- Circular motion: This extends the study of circular motion to both horizontal and vertical planes, covering concepts like angular speed, centripetal force, and how they apply in different scenarios, such as banked curves.
- Hooke's law: This covers the behavior of elastic strings and springs, including calculations involving elastic potential energy and the modulus of elasticity.
- Linear motion under a variable force: This topic uses differential equations to model and solve problems involving motion where the force is not constant.
- Momentum: This covers the principle of conservation of linear momentum, as well as the concept of the coefficient of restitution and its application to collisions (direct and oblique).
Further Probability and Statistics
- Probability density function (PDF): Defining and calculating probabilities, as well as finding the mode, median, and percentiles for a variety of continuous distributions.
- Cumulative distribution function (CDF): Calculating the CDF from the PDF and using it to find probabilities.
- Expectation and variance: Calculating the expectation
E(X)cap E open paren cap X close paren𝐸(𝑋) and variance Var(X)cap V a r open paren cap X close paren𝑉𝑎𝑟(𝑋) for both discrete and continuous random variables and their functions. - Uniform distribution: A new continuous distribution where the probability is constant over a given interval.
- Exponential distribution: A continuous distribution used to model waiting times, which is covered by boards like Cambridge . .