# St Paul's School for Girls

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### A Level Pure Maths

 Unit Title Estimated hours 1 Proof: Examples including proof by deduction* and proof by contradiction 3 2 Algebraic and partial fractions a Simplifying algebraic fractions 2 b Partial fractions 3 3 Functions and modelling a Modulus function 2 b Composite and inverse functions 3 c Transformations 3 d Modelling with functions* 2 *examples may be Trigonometric, exponential, reciprocal etc. 4 Series and sequences a Arithmetic and geometric progressions (proofs of ‘sum formulae’) 4 b Sigma notation 2 c Recurrence and iterations 3 5 The binomial theorem a Expanding (a + bx)n for rational n; knowledge of range of validity 4 b Expansion of functions by first using partial fractions 3 6 Trigonometry a Radians (exact values), arcs and sectors 4 b Small angles 2 c Secant, cosecant and cotangent (definitions, identities and graphs); Inverse trigonometrical functions; Inverse trigonometrical functions 3 d Compound* and double (and half) angle formulae 6 *geometric proofs expected e R cos (x ± α) or R sin (x ± α) 3 f Proving trigonometric identities 4 g Solving problems in context (e.g. mechanics) 2 7 Parametric equations a Definition and converting between parametric and Cartesian forms 3 b Curve sketching and modelling 2

 Unit Title Estimated hours 8 Differentiation a Differentiating sin x and cos x from first principles 2 b Differentiating exponentials and logarithms 3 c Differentiating products, quotients, implicit and parametric functions. 6 d Second derivatives (rates of change of gradient, inflections) 2 e Rates of change problems* (including growth and kinematics) 3 *see Integration (part 2) – Differential equations 9 Numerical methods* a Location of roots 1 b Solving by iterative methods (knowledge of ‘staircase and cobweb’ diagrams) 3 c Newton-Raphson method 2 d Problem solving 2 *See Integration (part 2) for the trapezium rule 10 Integration (part 1) a Integrating xn (including when n = –1), exponentials and trigonometric functions. Integrating functions defined parametrically. 4 b Using the reverse of differentiation, and using trigonometric identities to manipulate integrals 5 11 Integration (part 2) a Integration by substitution 4 b Integration by parts 3 c Use of partial fractions 2 d Areas under graphs or between two curves, including understanding the area is the limit of a sum (using sigma notation). Areas under curves expressed parametrically 4 e The trapezium rule 2 f Differential equations (including knowledge of the family of solution curves) 4 12 Vectors (3D): Use of vectors in three dimensions; knowledge of column vectors and i, j and k unit vectors 5 120 hours

### A Level Applied Content

 Unit Title Estimated hours Section A – Statistics 1 Regression and correlation a Change of variable 2 b Correlation coefficients Statistical hypothesis testing for zero correlation 5 2 Probability a Using set notation for probability Conditional probability 5 b Questioning assumptions in probability 2 3 The Normal distribution a Understand and use the Normal distribution 5 b Use the Normal distribution as an approximation to the binomial distribution Selecting the appropriate distribution 5 c Statistical hypothesis testing for the mean of the Normal distribution 6 30 hours Section B – Mechanics 4 Moments: Forces’ turning effect 5 5 Forces at any angle a Resolving forces 3 b Friction forces (including coefficient of friction µ) 3 6 Applications of kinematics: Projectiles 5 7 Applications of forces a Equilibrium and statics of a particle (including ladder problems) 4 b Dynamics of a particle 4 8 Further kinematics a Constant acceleration (equations of motion in 2D; the i, j system) 3 b Variable acceleration (use of calculus and finding vectors  and  at a given time) 3 30 hours