Further Mathematics (Year 13)
Exam Board
Edexcel
What will you study?
Complex numbers
• Understand de Moivre’s theorem and use it to find multiple angle formulae and sums of series.
• Know and use the definition eiθ = cos θ + i sin θ and the form z = reiθ.
• Find the n distinct nth roots of reiθ for r ≠ 0 and know that they form the vertices of a regular n-gon in the Argand diagram.
• Use complex roots of unity to solve geometric problems.
Series
• Understand and use the method of differences for summation of series including use of partial fractions.
• Find the Maclaurin series of a function including the general term.
• Recognise and use the Maclaurin series for ex, ln(1 + x), sin x, cos x and (1 + x)n, and be aware of the range of values of x for which they are valid (proof not required).
Methods in calculus
• Evaluate improper integrals where either the integrand is undefined at a value in the range of integration, or the range of integration extends to infinity.
• Understand and evaluate the mean value of a function.
• Integrate using partial fractions.
• Differentiate inverse trigonometric functions.
• Integrate functions of the form (𝑎2 −𝑥2 ) −12 (a2−x2)−12 and (𝑎2 −𝑥2 ) −1 (a2−x2)−1and be able to choose trigonometric substitutions to integrate associated functions.
Geometric and negative binomial distributions
• know when the geometric distribution is a suitable model and be able to apply it to a problem in context.
• know the necessary assumptions for the geometric distribution.
• be able to explain the meaning of the parameter p.
• be able to use the formulae for the mean and variance of the geometric distribution in a variety of problems.
The Central Limit Theorem
be able to quote the Central Limit Theorem.
• know the conditions for the use of the Central Limit Theorem.
• recognise, from problems in context, where the Central Limit Theorem is required.
Elastic strings and springs
be able to investigate the ability of strings to stretch and springs to stretch and compress.
be able to define the modulus of elasticity (λ), natural length (a) and extension (x).
be able to use the above definitions to work out the tension in a stretched string or a stretched/compressed spring i.e. use Hooke’s Law, T = λx/a ;
be able to derive the elastic potential energy (EPE) from Hooke’s Law by applying the work done in stretching a string/spring. i.e. EPE = (λx^2)/2a
Volumes of revolution
•be able to derive formulae for and calculate volumes of revolution about both the x and y-axes.
•be able to find volumes of revolution for functions given in parametric form.
Polar coordintes
•Understand and be able to use polar coordinates and be able to convert between polar and Cartesian coordinates.
•Know how to sketch standard polar curves.
•Be able to find tangents parallel and perpendicular to the initial line.
•Be able to find (compound) areas under polar graphs using the formula 1/2 ∫𝑟2 d𝜃.
Hyperbolic functions
•Understand the definitions of hyperbolic functions sinh 𝑥, cosh 𝑥 and tanh 𝑥, including their domains and ranges, and be able to sketch their graphs.
•Differentiate and integrate hyperbolic functions.
•Understand and be able to use the definitions of the inverse hyperbolic functions and their domains and ranges.
•Derive and use the logarithmic forms of the inverse hyperbolic functions.
•Integrate functions of the form (𝑥2 +𝑎2)−1/2 x2+a2)−1/2 , (𝑥2 −𝑎2)−1/2 (x2−a2)−1/2and be able to choose substitutions to integrate associated functions.
Probability generating functions
be able to use the definition of a probability generating function (pgf) to find the pgf of a given probability distribution.
be able to check the validity of a given pgf;
know how to use a given pgf to find the mean and variance of the probability distribution.
know how to use the power series form of a pgf to find individual probabilities.
be able to use the pgf of a discrete random variable X to find the pgf of a closely related random variable.
Elastic collisions in two dimensions
•Be able to solve problems of the following types involving elastic impacts:
a) successive collisions between pairs of spheres (horizontal motion).
b) bouncing ball (off a horizontal elastic plane);
c) successive collisions including two spheres and sphere against a wall;
d) determination of number of collisions or deriving the possible range of 𝑒.
Methods in differential equations
•Know how to deal with infinity as a limit of a definite integral.
•Be able to integrate functions across limits which include values when the function is undefined i.e. deal with discontinuous integrands.
•Understand and be able to evaluate the mean value of a function.
•Be able to integrate functions which can be split into partial fractions up to denominators with quadratic factors.
Modelling with differential equations
•Be able to use differential equations in modelling in kinematics and in other contexts.
•Be able to solve the equation for simple harmonic motion 𝑥 ¨ =−𝜔 2 𝑥 x¨=−ω2x and relate the solution to the motion.
•Be able to model damped oscillations using second order differential equations and interpret their solutions.
Quality of tests and estimators
•be able to state a definition of a Type I error and explain in the context of the question.
•be able to state a definition of a Type II error and explain in the context of the question.
•be able to use conditional probability with standard probability distributions to calculate the value of a Type II error.
Useful tips and resources
Pearsons text book.
Gold, silver and bronze papers.
End of unit tests, pearson.
Gold, silver and bronze papers.
End of unit tests, pearson.
What super curricular activities can KS3 students engage with at school for your subject?
Hans Woyda Maths Competition. Autumn term.
A-Level Wider Reading List:
Does God Play Dice by Ian Stewart.
Chaos by James Gleick.
The Codebook by Simon Singh.
The Mathematics of Ciphers by S.C. Coutinho.
In Code by Sara Flannery.
History of Mathematics.
A History of Mathematics by Carl B. Boyer.