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# Summary of the A-Level Specification

## Summary of the A-Level Specification

The complete A-Level Syllabus can be found on the Edexcel website; below is an abridged version to facilitate effective use of the site

Note that the AS syllabus appears to be a subset of the A-Level syllabus so I have highlighted in bold the material that uniquely belongs to A-Level and not AS.

Additionally, not all topics have yet been fully cross-referenced in the table below

### Pure Maths

Pure maths is split into ten distinct topics. (The additional topic is numerical methods)

#### Proof

 Description Links Proof by Deduction proof by deduction Proof By Exhaustion Disproof by Counterexample Proof by Contradiction

#### Algebra and functions

 Description Links Laws of indices surds manipulate surds, including rationalisation surds quadratics, graphs, discriminant, complete the square, solve quadratics, disguised quadratics quadratics disguised quadratics simultaneous equations of two variables, including one quadratic Simultaneous Eq linear and quadratic inequalities, interpret graphically. Express answers using 'and', 'or' OR through set notation. inequalities Manipulate polynomials, expand brackets, collect terms, factorise, algebraic division, factor theorem algebraic division factor theorem Sketch graphs of polynomials and reciprocals, including horizontal and vertical asymptotes; use graphs to solve equations. Understand proportionality. sketching 1 Understand the effect of simple transformations (a single stretch or translation) transformations Simplify Rational expressions The modulus of linear functions modulus function Composite Functions, inverses and their graphs composite functions Composites of simple transformations Partial fractions partial fractions Use of functions for modelling

#### Coordinate geometry

 Description Links equation of straight lines, in multiple forms. Gradient of parallel and perpendicular lines. Use straight line models straight lines equation of a circle, in geometric and cartesian form. complete square to determine centre and radius, use angle in a semi circle id a right angle. perpendicular  from centre bisects a chord. radius is perpendicular to tangent circle equation Parametric curves and conversion between parametric and Cartesian parametric curves Modelling with parametric curves

#### Sequences and series

 Description Links Understand and use the binomial expansion for positive integers, Extend to any rational $n$ and have awareness of radius of convergence binomial expansion Manipulate sequences, including those generated by iterative formulae. Understand and use sigma notation summation notation Understand and use arithmetic sequences arithmetic sequences Understand and use geometric sequences geometric sequences Use sequences and series in modelling

#### Trigonometry

 Description Links understand, use definitions of sin, cos, tan for all arguments; sine and cosine rules; area of a triangle intro trig use sin, cos, tan functions, graphs, symmetries and periodicity intro trig use identities $\tan (x)= \frac{\sin(x)}{\cos(x)}$ and Pythagorean identity Pythagorean id Solve trigonometric equations, in a given interval, including quadratic equations in trig functions, and linear combinations of the unknown. Radians Small angle approximations small angle approx Knowledge of exact values Trig values Reciprocal trig functions Trig identities, double angle formula, compound angle formula, R-alpha method, Pythagorean Identities with reciprocal functions double angle formula compound angle formula r-$\alpha$ method Trig Proofs Contextual problems

#### Exponentials and Logarithms

 Description Links know and use graph of the function $a^x$, where $a\gt 0$, know and use the function $e^x$ and its graph exponentials Know that the gradient of $e^{kx}$ is $ke^{kx}$ and hence understand why the exponential model is useful derivatives know and use $\log_ax$ as inverse of $a^x$ exp log inverse Know and use laws of logarithms. laws of logs Solve equations of the form $a^x = b$ laws of logs Use log graphs to establish parameters in the form  $y=ax^n$ and $y = kb^x$ understand and use exponential growth and decay models

#### Differentiation

 Differentiation Links understand the derivative as the gradient of the tangent to a graph at a general point, sketch gradient curves, find and interpret second derivatives, first principles differentiation first principles differentiate $x^n$ for rational $n$ and linear combinations differentiation apply differentiation to find gradients, tangents, normals, maxima, minima and stationary points. Identify where functions are increasing or decreasing. tangent line Differentiate $\sin(kx),\;\cos(kx),\;\tan(kx)$ $e^{kx},\ln{x}$ and related linear combinations tangent line Points of inflection, convex and concave curves tangent line Product, Quotient and Chain rules product rule quotient rule chain rule Implicit and parametric differentiation implicit parametric Construction of differential equations tangent line

#### Integration

 Integration Links Know and use the fundamental theorem of calculus ftoc Integrate $x^n$ (excluding $n=-1$) and related sums, differences and constant multiples intro to integration Evaluate definite integrals; use integral to find area under a curve intro to integration Integrate $\frac{1}{x}$, $e^{kx}$, $\sin(kx)$, $\cos(kx)$, $\tan(kx)$ and related functions, including use of identities trig integration Find the area between two curves Understand and use integration as the limit of a sum Integration by substitution substitution Integration by parts by parts integration using partial fractions partial fractions first order differential equations separation of vbls

#### Vectors

 Vectors Links use vectors in two dimensions intro vectors calculate magnitude and direction of a vector and convert between component form and magnitude / direction form intro vectors Add vectors, apply scalar multiplication and understand their geometric interpretations intro vectors Understand and use position vectors; calculate distance between two point represented by position vectors intro vectors