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A level Mathematics Content

A Level (Advanced Level) is usually studied over two years. The first year of an A Level is called AS (Advanced Subsidary) , the second year is called AS2.

DOE: ASs and A-Level Mathematics Content 2016 Formulae A level further maths

There are 19 components, labelled A to S :

(A ) Proof (B ) Algebra and functions (C ) Coordinate geometry in the (x,y) plane (D ) Sequences and series (E ) Trigonometry (F ) Exponentials and logarithms (G ) Differentiation (H ) Integration (I ) Numerical methods (J ) Vectors (K ) Statistical sampling (L ) Data presentation and interpretation (M )Probability (N )Statistical distributions (O ) Statistical hypothesis testing (P )Quantities and units in mechanics (Q ) Kinematics (R ) Forces and Newton’s laws (S ) Moments

There is overlap between AS Mathematics and the Scottish Higher Mathematics and Higher Applications of Mathematics courses
and also between AS2 Mathematics and the Scottish Advanced Higher Mathematicss and Advanced Higher Mathematics of Mechanics courses.

AS and A Level Content Overview

(A) Proof

A Proof

Mathematical proof Direct proof Proof by contradiction Proof by contrapositive Proof by induction

(B) Algebra and Functions

B Algebra and functions

Surds & Indices

Indices Surds Surds test

Simultaneous Equations

Simultaneous equations : graphical method Simultaneous equations : solving by elimination Simultaneous equations : Vedic method

Algebraic Manipulation

Factors - removing brackets Foil Factorising Difference of two squares Equations Inequalities Changing the subject of the formula

Completing the Square

Completing the square - overview Completing the square - general method Completing the square - solving quadratic equations Completing the square - usefulness for quadratics

Polynomials

Coefficients etc. Nested form. Dividing polynomials The factor theorem The remainder theorem
Synthetic division Dividing by ax+b Solving polynomial equations Sketching polynomial equations

Algebraic Fractions

Algebraic fractions Algebraic fraction operations

Functions and Graphs

Functions

Functions Using functions Set notation Domain, codomain & range One to one correspondence Inverse function Inverse function - algebraic method Inverse function - graphical method composite functions Simple graphs of functions Reflections, translations, etc

Graphs

Simple graphs of functions Reflections, translations, etc Sketching polynomial equations Graphing Exponential and Logarithmic functions Logarithmic - linear scales on graphs Log - log graphs Excel -Exponential / log functions - graphs

Partial Fractions

Rational fractions Partial fractions Distinct linear factors Repeated linear factors Irreducible quadratic factor

(C) Coordinate geometry in the (x,y) plane

C Coords

The Straight Line

General

y - b = m(x-a) general equation of a straight line gradient = tan θ distance formula midpoint formula perpendicular lines collinearity

Straight lines and circles

Altitudes Centroid Circumcentre Circumcircle Concurrency Median Orthocentre Perpendicular bisector

The Circle

Circles

Radius, Diameter and Circumference Area Arcs and Sectors Chords and things Angle in a semi circle
The circle - higher course examples equation of circle : x² + y² = r²

Equation of circle : given centre, radius Equation of circle : line joining diameter General equation of circle Intersection of line & circle Tangents to a circle Equation of a circle from chords Touching circles

Parametrics

Parametric equations Parametric equations:circles Parametric constraint equations Differentiating parametric Second Derivative: parametric equations Uses of parametric equations Equations of motion Projectile Motion Horizontal Range

(D) Sequences and series

D Sequences and Series

Sequences

nth term of sequence Recurrence relations Divergence / convergence Linear recurrence relations Formula for linear recurrence relations Solving recurrence relations Special sequences

Recurrence relations Fixed points Arithmetic sequences Arithmetic series n^th term of arithmetic sequence Sum to n terms of an arithmetic sequence Geometric sequences Geometric series Sum to n terms of a geometric sequence Infinite series Sum to infinity - geometric series Sum first n natural numbers Sigma notation - Rules Common series - sigma notation

Power Series

Power series D'alembert's ratio test Absolute convergence Fibonacci series Centre of convergence Taylor's series Maclaurin's Series Maclaurin expansion

(E) Trigonometry

E Trigonometry

Triangle sides Tan Sin Cos Which ratio ?
Sine Rule - derivation Sine Rule- Example Cosine Rule - derivation Cosine Rule- Examples Area of a triangle - derivation Area of a triangle - example Bearings with trigonometry Trigonometric identities Trigonometric Graphs Amplitude and Period

Algebra - Solving Trig equations Amplitude and Period Sketching trig graphs Compound angle formulae - Examples

Radians and Exact Values

Radians Converting degrees to Radians Exact values Radians & Degrees Using exact values : Radians & Degrees Conversion table

Wave Theory

Adding two waves Subtracting two waves Express in the form cos(x+a) Express in the form sin(x-a) Express in the form sin(x+a)

Multiple Angles Maximum / Miminum Value Sketching wave functions Wave Theory- Solving equations

Formulae

Trig refresher Trigonometric identities sec, cosec & cot
Compound formulae Double Angle Half Angle Product formulae Multiple Angle Formulae

Deriving formulae

Deriving addition angle formulae Deriving double angle formulae Deriving multiple angle formulae Deriving half angle formulae

(F) Exponentials and Logarithms

F Logs and Exponentials

Exponentials

Exponentials Growth functions Decay functions

Logarithms

Logarithms Logarithms - converting bases logarithm laws Log and Ln Logarithmic equations

Graphs

Graphing Exponential and Logarithmic functions Logarithmic - linear scales on graphs Log - log graphs Excel -Exponential / log functions - graphs

Calculus

(G) Differentiation

G Differentiation

First Principles

First principles x^n ax^n
Derivatives of sums (x + a)^n (ax + b)^n
The chain rule The product rule
The quotient rule Trig functions

Main Differentiation

Differentiation Gradients Notation
xⁿ axⁿ Derivatives of sums
Differentiating Sin and Cos functions Differentiating (x + a)ⁿ Differentiating (ax + b)ⁿ The chain rule Product rule. Quotient rule

Stationary Points and Tangents

Increasing / decreasing functions Stationary points finding stationary points
maxima and minima closed intervals Tangents- gradients Tangents- equations

Optimisation and Curve Sketching

Applications curve sketching graphing derived functions Optimisation problems Newton's equations of motion Rectilinear motion

seca coseca cota Higher Derivatives Formulae for differentiation

Rules

Exponentials & Logs Differentiating inverse functions
Differentiating inverse trig functions Inverse trig functions:Recap
Implicit & explicit functions Logarithmic functions

Parametric Differentiation

Differentiating parametric Second Derivative: parametric equations

Differential Equations

Differential equations First-order differential equations Second-order differential equations Applications of differential equations

Applications of Calculus

Rectilinear motion Distance, velocity & acceleration Extrema of functions Approximating roots Uses of parametric equations Equations of motion

(H) Integration

H Integration

Basic Integration

Integral calculus indefinite integrals basic rules of integration Integration and the area function Definite integrals Fundamental theorem of calculus Integrating trig functions Indefinite integral of (ax+b)ⁿ Differential equations Standard integrals e^x and 1/x
Infinite integrals (discontinuities) sec²x
Integrating Inverse trig functions Integration of rational functions

Integration by substitution. Integration by parts

(I) Numerical methods

I Numerical Methods

Area under curves

Areas enclosed by graph and x-axis Area between two graphs Formula for area between two graphs Areas of graphs Areas under curves Area between curve and the y-axis Volumes of revolution

Application of summation formulae

Standard series Examples Iteration Iteration - graphical method First order process Second order process Rule of false position Newton-Raphson iteration

(J) Vectors

J Vectors

Basic Vectors

Scalar and vector quantities Components Magnitude Position vectors Unit vector Zero vector Equal vectors Adding vectors Subtracting vectors Perpendicular vectors

Other Vectors

Collinearity Angle between vectors Scalar product Section formula Linear motion with bearings Vectors and bearings Relative Motion
Vectors and forces

Statistics

(K) Statistical sampling

K statistical sampling

(L) Data presentation and interpretation

L Data presentation and intepretation

(M) Probability

M Probability

(N) Statistical distributions

N Statistical distribution

(O) Statistical hypothesis testing

O Statistical hypothesis testing

An Introduction to SQA Statistics Higher Unit Stats Award : R Studio guide
Refresher : National 5 Apps Statistics
Scatter diagrams Line of best fit Residual
Types of Data Skewness

Statistical Tests

Sets Venn Diagrams Pearson’s product-moment correlation coefficient
Normal Distribution Curve ( Bell Curve) t-tests Paired t-tests
Hypothesis Testing Confidence Intervals z-tests

Probability

Probability Mutually Exclusive Events
Addition Law Multiplication Law
Tree Diagrams Two Way Tables

Factorials Permutations Combinations
Pascal’s Triangle. Binomial Co-efficients Binomial Co-efficient equations Binomial Expansions
Binomial Theorem Probability and the Binomial Theorem Binomial Expansions and e

Mechanics

(P) Quantities and units in mechanics

P Quantities and units in mechanics

(Q) Kinematics

Q Kinematics

(R) Forces and Newton’s laws

R Forces and Newton's laws

(S) Moments

S Moments

Reading graphs Areas of graphs Deriving Newton's equations of motion Equations of motion Projectile Motion Horizontal Range

Parametric equations Parametric equations:circles Parametric constraint equations Differentiating parametric Second Derivative: parametric equations Uses of parametric equations

Linear motion with bearings Vectors and bearings Relative Motion Vectors and forces Law of Statics Pulleys Single pulleys, acceleration and thrust

Newton's Laws of Motion Closed Triangle of Forces in Equilibrium Deriving Lami's Theorem

Objects on inclined planes Coefficient of friction

Turning moments Static Rigid Bodies Couples Centre of Mass

Books

A selection of resources available at Amazon

AS Mathematics GCP: A-Level Maths Edexcel A-Level Further Maths
Understanding Mechanics Engineering Mathematics Engineering Mathematics Advanced Higher Physics Theory Advanced Higher Physics Investigations Companion Advanced Higher Physics - Questions and Solutions

Maths Mutt at Amazon

New Advanced Higher Mathematics
Revision Notes Differentiation
By First Principles Quick Forty
Integration and Differential Equations

MM Amazon Store UK school subjects Maths Mutt Publications

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