Further Mathematics (Year 12)
Exam Board
Edexcel
What will you study?
Complex numbers
• Be able to solve any quadratic equation with real coefficients.
• Be able to add, subtract and multiply complex numbers in the form 𝑥+i𝑦 with 𝑥 and 𝑦 real.
• Understand and use the terms ‘real part’ and ‘imaginary part’.
• Understand and use the complex conjugate of a complex number.
• Be able to divide two complex numbers by using the complex conjugate of the denominator.
• Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs.
• Be able to solve cubic or quartic equations with real coefficients.
Argand diagrams
• Be able to show complex numbers on an Argand diagram.
• Find the modulus and the argument for a complex number.
• Write a complex number in a modulus argument form.
• Represent loci on an Argand diagram
• Be able to construct and interpret simple loci in the Argand diagram such as |𝑧−𝑎|>𝑟|z−a|>r and arg(𝑧−𝑎)=𝜃arg(z−a)=θ.
• Represent regions on an Argand diagram.
• Understand the geometrical effects of adding and subtracting two complex numbers.
• Understand the meanings of the modulus and argument of a complex number
• Know that |z1z2| = |z1||z2|
Series
• Be able to use sigma notation (Σ)
• Be able to use the standard series formulae for Σr, Σr², Σr³ to find the sums of series.
Roots of polynomials
• Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations.
• Be able to form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).
Proof by induction
• Construct proofs using mathematical induction
Contexts include sums of series, divisibility and powers of matrices.
• Be able to obtain a proof for the summation of a series, using induction.
• Be able to use proof by induction to prove that an expression is divisible by a certain integer.
• Be able to use mathematical induction to prove general statements involving matrix multiplication.
Matrices
•Add, subtract and multiply conformable matrices. Multiply a matrix by a scalar.
•Understand and use zero and identity matrices.
•Use matrices to represent linear transformations in 2D.
Successive transformations.
Single transformations in 3D.
•Find invariant points and lines for a linear transformation.
•Calculate determinants of 2×2 and 3×3 matrices and interpret as scale factors, including the effect on orientation.
•Understand and use singular and non-singular matrices.
•Properties of inverse matrices.
Calculate and use the inverse of non-singular 2×2 matrices and 3×3 matrices.
•Solve three linear simultaneous equations in three variables by use of the inverse matrix.
•Interpret geometrically the solution and failure of solution of three simultaneous linear equations.
Linear transformation
•be able to use matrices to represent 2D rotations, reflections, enlargements, and translations.
•understand and use zero and identity matrices.
•be able to use matrix products to represent combinations of transformations.
•be able to use matrices to represent linear transformations in three dimensions.
•be able to use inverse matrices to reverse the effect of a linear transformation.
•be able to use the determinant of a matrix to determine the area scale factor of a transformation.
•be able to find invariant points and lines for a linear transformation.
Vectors
•Understand and use the vector and Cartesian forms of an equation of a straight line in 3D
•Understand and use the vector and Cartesian forms of the equation of a plane.
•Calculate the scalar product and use it to express the equation of a plane, and to calculate the angle between two lines, the angle between two planes and the angle between a line and a plane.
•Check whether vectors are perpendicular by using the scalar product.
•Find the intersection between a line and a plane
Calculate the perpendicular distance between two lines, from a point to a line and from a point to a plane
Discrete probability distributions
be able to calculate the mean and variance of discrete probability distributions using
E(X)=μ=∑▒〖xP(X=x)〗 and Var(X)=σ^2=∑▒〖x^2 P(X=x)-μ^2 〗;
know how to find the expectation of a function of a random variable.
Discrete probability distributions
be able to calculate the mean and variance of discrete probability distributions using
E(X)=μ=∑▒〖xP(X=x)〗 and Var(X)=σ^2=∑▒〖x^2 P(X=x)-μ^2 〗;
know how to find the expectation of a function of a random variable.
Momentum and impulse
•Intuitive idea of momentum and S.I. units
•Kinematics – constant acceleration formulae
Work, energy and power
•understand the derivation, units and definitions of work and energy.
•be able to define kinetic energy (KE).
•understand that work done on a body moving in a horizontal plane is the change in kinetic energy.
•understand that power in watts is the rate of doing work.
•be able to calculate the power (P) of a vehicle with a tractive (driving) force 𝐹, moving with velocity 𝑣.
•be able to use the formula 𝑃=𝐹𝑣 in problem solving.
Hypothesis testing
• understand and be able to apply the language of statistical hypothesis testing.
• be able to carry out hypothesis tests to test for the mean of a Poisson distribution.
Chi squared tests
know and be able to use the process of a goodness of fit test.
know how to find the number of degrees of freedom of the expected values, including when one or more parameters are estimated from the data.
be able to use ∑_(i=1)^n▒(O_i-E_i )^2/E_i as an approximate 2 statistic.
be able to apply goodness of fit tests to include the discrete uniform, binomial, Poisson and geometric distributions.
be able to show frequencies by means of a contingency table.
know how to obtain p-values from calculators.
be able to use tables to find critical values.
Elastic collisions in one dimension
• be able to express the ‘compressibility’, ‘bounciness’ or ‘elasticity’ of an object by a value called the coefficient of restitution (𝑒);
• know that 0≤𝑒≤1 [and that 𝑒 = 0 means inelastic and 𝑒 = 1 means perfectly elastic];
• know and be able to use Newton’s (experimental) law of restitution for direct impacts of elastic spheres.
• be able to calculate the change in kinetic energy due to an impact.
Volumes of revolution
• Be able to derive formulae for and calculate volumes of revolution about both the 𝑥 and 𝑦-axes.
• Subtracting one volume from the other.
• Finding the volume of shapes rotated around a line that is parallel to one of the axes.
Useful tips and resources
Pearson text book.
Gold, silver bronze practice papers.
End of unit tests (Pearson).Mr Carter Maths
• Dr Frost Maths
• TL Maths TLMaths – A-Level Maths
• Desmos
• Mr Barton Maths
• AMSP
• Physics and Maths Tutor Edexcel A-level Maths Revision – PMT
• Maths Genie
What super curricular activities can KS3 students engage with at school?
KS5 students will take part in the Hans Woyda, which is a highly regarded and fiercely competitive Mathematics competition held in London. Sixty-four schools, from both state and private schools, take part in the competition.