Quadratic Functions, Equations and Inequalities

• Solve and sketch any quadratic including hidden quadratics
• Completed the square form
• Solve quadratic inequalities
• Discriminant
• Simultaneous equations

Problem Solving and Proof

• Solving problems 
• Use mathematical symbols and notation including implication
• Formal methods of proof 

Polynomials

• Add subtract and multiply any two polynomials
• Factor theorem and algebraic division
• Know shape and features of graphs of polynomials

Graphs

• Transformations of graphs 
• Reciprocal graphs
• Direct proportion graphs
• Inequalities on graphs

Differentiation

• Differentiation from first principles
• Differentiate polynomials 
• Equation of tangents and normal to curves
• Increasing and decreasing functions
• Finding and classifying stationary points

Surds and Indices

• Structure of the number system
• Add, subtract and multiply surds
• Rationalise the denominator
• Use all laws of indices 
• Write numbers as a power of another 
• Solve problems using indices

Trigonometry

• Sine, cosine and sine rule for area
• Graphs of trigonometric functions
• Trig identities 
• Solve equations involving trig functions

Coordinate Geometry 

• Distance between two points and midpoint
• Equation of a line in a given format
• Equation of parallel and perpendicular lines
• Equation of a circle
• Equation of the tangent to the circle
• Simultaneous equations with circles

Binomial Expansion

• Binomial expansion to find terms 
• Use binomial coefficient to find unknowns
• Use binomial expansion when two binomials are being multiplied
• Use expansions to approximate

Differentiation

• Differentiation from first principles
• Differentiate polynomials 
• Equation of tangents and normal to curves
• Increasing and decreasing functions
• Finding and classifying stationary points

Exponentials and Logarithms

• Sketch exponential functions
• Derivative of e^kx
• Use laws of logarithms
• Use logarithms to solve exponential equations
• Sketch logarithmic graphs 
• Modelling with exponentials
• Logarithms to transform graphs 

Data Collection

• Data handling cycle
• Different types of data 
• Types of sampling 

Data Processing and Interpretation

• Histograms
• Cumulative frequency 
• Variance and standard deviation
• Scatter diagrams
• Outliers

Binomial Expansion

Integration 

• Integrate functions with terms of the form ax^n
• Find constant of integration
• Definite integration for area under a curve
• Area between a curve and a line

Vectors

• Change between column vectors and magnitude and direction
• Add and subtract vectors 
• Geometric problems with vectors
• Vector proof

Kinematics and Variable Acceleration

Probability

• Calculate probabilities 
• Show two events are independent
• Use tree diagrams and Venn diagrams
• Discrete probability distributions

Binomial Distribution

• Formula for number of combinations
• Properties of Binomial distribution
• Calculate probabilities with Binomial distribution 

Hypothesis Testing

• One and two tailed hypothesis tests with Binomial distribution

Large Data Set

Kinematics and Variable Acceleration 

• Distance-time, displacement-time and velocity-time graphs
• Derive constant acceleration formulae
• Use constant acceleration formulae
• Simultaneous equations and the constant acceleration formulae
• Effect of assumptions
• Link between acceleration, velocity and displacement
• Solve problems with variable acceleration

Forces and Motion

• Force diagrams
• Newton’s laws
• Types of forces
• Solve problems with Newton’s laws
• Modelling assumptions

Functions

• Types of functions
• Graphs of functions including discontinuous
• Domain and range
• Composite and inverse functions 
• Modulus function
• Solve equations/ inequalities with the modulus function
• Graph transformations

Differentiation

• Differentiate ln⁡x, sin⁡x, cos⁡x, tan⁡x
• Chain rule
• Product and Quotient rule
• Implicit differentiation 
• Differentiate reciprocal trig functions
• Convex and concave curves
• Connected rates of change 

Trig Identities

• Compound angle formulae
• Double angle formulae
• New identities and proof
• Form Rcos(x+c) and Rsin(x+d) 

Integration

• Integrate Exponential and Trigonometric functions
• Area between two curves
• Reverse chain rule
• Integration by substitution
• Integrate with partial fractions
• Integration by parts
• Integration and trig identities

Proof

• Proof by exhaustion
• Direct proof
• Proof by contradiction

Trigonometry 

• Radians
• Graphs of Trig functions
• Area and arc length of a sector
• Are of a segment
• Small angle approximations

Trig Functions

• Reciprocal trig functions and their graphs
• Derive new identities
• Inverse trig functions
• Solve trig equations with reciprocal trig functions

Further Algebra

• Factor theorem with (ax+b)
• Remainders with algebraic division
• Partial fractions
• Expand (ax+b)^n 

Sequences and Series

• Different types of sequences and series
• Convergence or divergence
• Sigma notation
• Arithmetic series
• Geometric series 

Parametric Equations

• Graphs of parametric equations
• Convert to Cartesian form 
• Differentiate and find gradients
• Integrate and find areas

Differential Equations

• Separation of variables    
• General solution
• Particular solution
• Forming differential equations

Vectors and Kinematics

• 3D vectors, magnitude and direction
• Use vectors in kinematics (suvat and F=ma)

Forces. Motion and Friction

• Resolve forces on inclined plane
• Friction
• Solve problems involving friction and inclined planes

Moments

• Moment on uniform rods and lamina
• Equilibrium and moments

Projectiles

• Resolve vertically and horizontally
• Solve projectile problems
• Equation of trajectory 

Numerical Methods

• Change of sign to find a root
• Newton Raphson
• Staircase and cobweb diagrams
• Using rectangles to estimate area
• Trapezium rule 

Probability 

• Set notation to describe probability 
• Conditional probability formula
• Solve problems with conditional probability 

Stats Distributions

• Normal distribution finding probabilities
• Reverse normal distribution
• Standardise to find unknowns
• Approximate binomial distribution 

Hypothesis Testing

• Normal distribution when using a sample
• Hypothesis testing with normal distribution
• Product moment correlation coefficient 

Revision for Exams and Gap Fill 

Examinations

Complex Numbers

• Extend number system 
• Solve quadratics with complex roots
• Complex conjugate
• Argand diagram
• Modulus argument form
• Loci of complex numbers

Sequences and Series

• Standard results for sum of integers, squares and cubes
• Method of differences
• Maclaurin series standard results 

Proof by Induction

• Proof by induction with series
• Proof by induction with matrices
• Proof by induction with divisibility 

Polar Coordinates

• Convert between polar and cartesian coordinates
• Sketch polar curves
• Tangents at the poles

Vectors

• Vector and Cartesian form of a 3D straight line
• Intersection point between two lines
• Angle between two lines
• Parallel and perpendicular lines
• Shortest distance

Hyperbolic Functions

• Exponential form of hyperbolics
• Graphs, domains and ranges of hyperbolics
• Inverse hyperbolics
• Solve equations with hyperbolics

Conics

• Equations and graphs of conic sections
• Equation of tangents
• Intersection of lines with conic curves
• Transformations

Matrices and Transformations

• Multiplication of matrices
• Associativity and commutativity 
• Determinant of a matrix
• Inverse of 2x2 matrices
• Matrix transformations
• Geometric interpretations
• Invariant points and lines  

Rational Functions

• Sketch graphs of form (ax+b)/(cx+d)  or (ax^2+bx+c)/(dx^2+ex+f)
• Inequalities using graph
• Inequalities creating cubics/quartics
• Possible value of a function and stationary point

Roots of Polynomials

• Factorise and solve polynomials with complex roots
• Relationship between roots and coefficients
• Linear transformation of roots
• Complex roots with real coefficients

Graphs

• Language and terminology of graphs
• Euler’s formula for connected planar graphs
• Adjacency matrices

Networks

• Terminology of networks
• Kruskal’s algorithm 
• Primm’s algorithm
• Route inspection problems
• Chinese postman problem
• Nearest neighbour algorithm 

Linear Programming

• Create linear programming problem
• Graph inequalities
• Solve optimisation problems

Critical Path Analysis

• Construct precedence networks
• Latest start time and earliest finish times
• Refine models and effect 

Network Flows

• Interpret flow problems
• Terminology 
• Find cuts 
• Max flow-minimum cut theorem
• Supersource and supersink

Game Theory 

• Construct payoff matrices
• Play safe strategies
• Value of a game
• Stable solution
• Dominated strategies
• Optimal mixed strategies

Forces, Motion and Friction 

• Force diagrams and Newton’s laws of motion
• Equilibrium
• Friction
• Vectors in two dimensions

Kinematics

• Derive constant acceleration formulae
• Use constant acceleration formulae 
• Effect of assumptions
• Solve problems with variable acceleration

Work, Energy, Power and Hooke's Law

• Work done
• Kinetic and gravitational potential energy 
• Conservation of energy 
• Work done by a variable force
• Hooke’s law

Momentum and Collisions

• Momentum and impulse
• Impulse of a variable force
• Conservation of momentum
• Newton’s experimental law

Binary Operations

• Construct payoff matrices
• Play safe strategies
• Value of a game
• Stable solution
• Dominated strategies
• Optimal mixed strategies

Further Calculus

• Proof of volume of a revolution
• Find volumes about x and y axes
• Mean value of a function 

Vectors 

• Vector product and application
• Equation of a plane in vector and Cartesian format 
• Ways planes can intersect 
• Angles between lines and planes

Polar Coordinates

• Area enclosed by polar curves

Momentum and collisions

• Momentum and impulse
• Impulse of a variable force
• Conservation of momentum
• Newton’s experimental law

Dimensional Analysis

• Dimensional analysis notation
• Validity of formulae
• Predict formulae

Circular Motion

• Forces in circular motion 
• Linear speed and angular speed
• Acceleration 
• Solve problems

Further Graphs and Inequalities

• Modulus reciprocal graphs and inequalities
• Draw reciprocal graphs with obliques asymptotes
• Sketch and solve equations and inequalities with y=|f(x)|

Conics

• Equation of conics after combinations of transformations
• Find combinations of transformations applied
• Parametric form of conics

Matrices

• Determinant and inverse of a 3x3 matrix
• Row/column operation
• 3 variable simultaneous equations
• Eigenvalues and eigenvectors

Complex Numbers

• De Moivre’s theorem 
• Exponential form of a complex number
• Roots of unity 
• Factorise z^n-a
• De Moivre’s theorem to derive trig identities

Hyperbolic Functions

• Inverse hyperbolic functions 
• Reciprocal hyperbolic functions
• Hyperbolic identities
• Differentiation of hyperbolics
• Integration of hyperbolics and inverse hyperbolics

Further Calculus

• Differentiate inverse trig functions
• Inverse trig and hyperbolic functions with substitution
• Partial fractions 
• Reduction formulae
• Length of an arc 
• Surface area of revolution

Series and Limits

• Maclaurin’s series
• L’Hopital’s rule 
• Improper integrals and limits

Differential Equations

• Integrating factor
• 2nd order differential equations 

Application of Differential Equations

• Simple harmonic motion 
• Damping 
• Coupled first order differential equations

Numerical Methods

• Mid-ordinate rule
• Simpson’s rule
• Euler’s method
• Euler’s improved method

Work, Energy and Power

Momentum and Collisions

Dimensional Analysis

Cicular Motion

• Problems with horizontal circular motion
• Variable speed
• Vertical circle
• Solve problems 

Moments and Couples

• Solve problems with moments
• Vector method 

Centre of Mass

• Centre of mass of particles
• Centre of mass of standard shapes and composite bodies
• Integration to find centre of mass
• Equilibrium, sliding and toppling

Graphs

Linear Programming

• Simplex algorithm 
• Interpret a simplex tableau

Critical Path Analysis

• Gantt Chart 
• Resource histogram
• Resource levelling

Network Flows

• Augment flows
• Upper and lower arc capacity problems
• Refine network flows 

Game Theory

• Optimal mixed strategies with simplex algorithm 
• Solve higher order games 

Groups

• Language of groups 
• Group axioms 
• Finite and infinite groups
• Legrange’s theorem 
• Isomorphisms

Revision and Gap Filling

Examinations