WJEC Level 3 Advanced GCE in Further Mathematics

Understand and use circular motion.

Understand and use the motion of a particle in a horizontal circle with uniform angular speed.

Understand and use the motion in a vertical circle.

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 for 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.

Solve any quadratic equation with real coefficients.

Solve cubic or quartic equations with real coefficients (given sufficient information to deduce at least one root for cubics or at least one complex root or quadratic factor for quartics).

Add, subtract, multiply and divide 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.

Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs

Equate the real and imaginary parts of a complex number.

Use and interpret Argand diagrams

Understand and use the Cartesian (algebraic) and modulus-argument (trigonometric) forms of a complex number.

Convert between the Cartesian form and modulus-argument form of a complex number.

Multiply and divide complex numbers in modulus-argument form.

Construct and interpret simple loci in an Argand diagram, such as |z − a| > r and arg(z − a) = θ.

Simple cases of transformations of lines and curves defined by w = f(z).

Find and use an integrating factor to solve differential equations of the form dy/dx + P(x)y = Q(x) and recognise when it is appropriate to do so.

Find both general and particular solutions to differential equations.

Use differential equations in modelling in a variety of contexts.

Solve differential equations of the form y″ + ay′ + by = 0, where a and b are constants, by using the auxiliary equation.

Solve differential equations of the form y″ + ay′ + by = f(x), where a and b are constants, by solving the homogenous case and adding a particular integral to the complementary function (in cases where f(x) is a polynomial, exponential or trigonometric function).

Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation.

Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled 1st order simultaneous equations and be able to solve them.

Use differential equations in modelling in kinematics.

Understand and use simple harmonic motion.

Differentiate and integrate vectors in component form with respect to a scalar variable.

Understand and use vector quantities including displacement, velocity, acceleration, force and momentum.

Understand and use the equilibrium of a single rigid body under the action of coplanar forces where the forces are not all parallel.

Understand and use confidence intervals:

Understand and use confidence limits for (a) the mean of a normal distribution with (i) known variance and (ii) unknown variance, (b) the difference between the means of two normal distributions whose variances are known.

Understand and use approximate confidence limits, given large samples, for a probability or a proportion.

Interpret results in practical contexts.

Understand and use correlation and linear regression:

Explore the relationships between several variables.

Calculate and interpret • Spearman’s rank correlation coefficient • Pearson’s product-moment correlation coefficient.

Calculate and interpret the coefficients for a least squares regression line in context; interpolation and extrapolation.

Understand and use the Chi-squared distribution:

Conduct goodness of fit test using ∑(O − E)²/E, or equivalent form, as an approximate χ² statistic (for use with categorical data).

Use χ² test to test for association in a contingency table and interpret results

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.

Understand and use partial fractions with denominators of the form (ax + b)(cx² + d).

Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations.

Form a polynomial equation whose roots are a transformation or a combination of the roots of a given polynomial equation.

Understand and use formulae for the sums of integers, squares and cubes and use these to sum other series.

Understand and use the method of differences for summation of series, including the use of partial fractions.

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.

Derive formulae for and calculate volumes of revolution.

Understand and evaluate the mean value of a function.

Integrate using partial fractions (extend to quadratic factors (ax² + c) in the denominator).

Differentiate inverse trigonometric functions.

Integrate functions of the form 1/√(a² − x²) and 1/(a² + x²) and be able to choose trigonometric substitutions to integrate associated functions.

Solve trigonometric equations.

Use the formulae for sin A ± sin B, cos A ± cos B and for sin x, cos x and tan x in terms of t, where t = tan ½x.

Find the general solution of trigonometric equations.

Understand and use the vector and Cartesian forms of an equation of a straight line in 3-D.

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.

Use the scalar product to check whether vectors are perpendicular.

Find the intersection of a line and a plane.

Calculate the perpendicular distance between two lines, from a point to a line and a point to a plane.

Solve problems involving light strings and springs obeying Hooke’s Law.

Understand and use work, energy and power.

Understand and use gravitational potential energy, kinetic energy, elastic energy.

Understand and use conservation of energy.

Understand and use the Work-energy Principle.

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 1/√(x² + a²) and 1/√(x² − a²), and be able to choose substitutions to integrate associated functions.

Understand and use tests for: (a) a specified mean of any distribution whose variance is estimated from a large sample. (b) difference of two means for two independent normal distributions with known variances. (c) a specified mean of a normal distribution with unknown variance.

Interpret results for these tests in context.

Non-parametric tests: Understand and use Mann-Whitney and Wilcoxon signed-rank tests, understanding appropriate test selection and interpreting the results in context.

Calculate determinants up to 3 x 3 matrices and interpret as scale factors, including the effect on orientation.

Calculate and use the inverse of non-singular 3 x 3 matrices.

Solve three linear simultaneous equations in three variables by use of the inverse matrix and by reduction to echelon form.

Understand and use the determinantal condition for the solution of simultaneous equations which have a unique solution.

Interpret geometrically the solution and failure of three simultaneous linear equations.

Add, subtract and multiply conformable matrices.

Multiply a matrix by a scalar.

Understand and use zero and identity matrices.

Understand and use the transpose of a 2 x 2 matrix.

Use matrices to represent • linear and non-linear transformations in 2-D, involving 2 x 2 and 3 x 3 matrices, • successive transformations, • single transformations in 3-D (3-D transformations confined to reflection in one of x = 0, y = 0, z = 0 or rotation about one of the coordinate axes).

Find invariant points and lines for linear and non-linear transformations.

Calculate determinants of 2 x 2 matrices.

Understand and use singular and non-singular matrices.

Understand and use properties of inverse matrices.

Calculate and use the inverse of non-singular 2 x 2 matrices.

Understand and use the centre of mass of a coplanar system of particles.

Understand and use the centre of mass of uniform laminae: triangles, rectangles, circles, semicircles, quarter-circles and composite shapes.

Solve problems involving simple cases of equilibrium of a plane lamina and/or a coplanar system of particles connected by light rods.

Understand and use the centre of mass of uniform rigid bodies and composite bodies.

Understand and use momentum and impulse in two dimensions, using vectors.

Understand and use momentum and impulse.

Understand and use conservation of momentum.

Understand and use Newton’s Experimental Law for (i) the direct impact of two bodies moving in the same straight line, (ii) the impact of a body moving at right-angles to a plane.

Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable

Understand and use mathematical language and syntax as set out in the content

Understand and use language and symbols associated with set theory, as set out in the content

Understand and use the definition of a function; domain and range of functions

Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics

Translate a situation in context into a mathematical model, making simplifying assumptions

Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the learner)

Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the learner)

Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate

Understand and use modelling assumptions

Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved

Construct extended arguments to solve problems presented in an unstructured form, including problems in context

Interpret and communicate solutions in the context of the original problem

Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle

Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems

Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates.

Sketch curves with r given as a function of θ, including the use of trigonometric functions.

Find the area enclosed by a polar curve.

Construct proofs using mathematical induction.

Contexts include sums of series, powers of matrices and divisibility.

Understand and use the mean and variance of linear combinations of independent random variables. ie. use of results: E(aX + b) = aE(X) + b; Var(aX + b) = a²Var(X); E(aX + bY) = aE(X) + bE(Y)

For independent X and Y, use the results: E(XY) = E(X) E(Y); Var(aX + bY) = a²Var(X) + b²Var(Y)

Probability: Discrete probability distributions. Find the mean and variance of simple discrete probability distributions.

Probability: Continuous probability distributions. Understand and use probability density and cumulative distribution functions and their relationships.

Find and use the median, quartiles and percentiles.

Find and use the mean, variance and standard deviation.

Understand and use the expected value of a function of a continuous random variable.

Statistical distributions: Poisson and exponential distributions. Find and use the mean and variance of a Poisson distribution and an exponential distribution.

Understand and use Poisson as an approximation to the binomial distribution.

Apply the result that the sum of independent Poisson random variables has a Poisson distribution.

Use of the exponential distribution as a model for intervals between events.

Form and solve simple equations of motion in which (i) acceleration is given as a function of time, displacement or velocity, (ii) velocity is given as a function of time or displacement.

Understand and use unbiased estimators:

Understand and use the variance criterion for choosing between unbiased estimators.

Understand and use unbiased estimators of a probability and of a population mean and their standard errors.

Understand and use an unbiased estimator of a population variance.

Understand and use the result that a linear combination of independent normally distributed random variables has a normal distribution.

Understand and use the fact that the distribution of the mean of a random sample from a normal distribution with known mean and variance is also normal.

Know and use the Central Limit Theorem:

Understand and use the fact that the distribution of the mean of a large random sample from any distribution with known mean and variance is approximately normally distributed.

This specification requires learners, during the course of their study, to: • develop skills relevant to exploring and analysing large data sets (these data must be real and sufficiently rich to enable the concepts and skills of data presentation and interpretation in the specification to be explored); • use technology such as spreadsheets or specialist statistical packages to explore data sets; • interpret real data presented in summary or graphical form; • use data to investigate questions arising in real contexts.

Learners should be able to demonstrate the ability to explore large data sets, and associated contexts, during their course of study to enable them to perform tasks, and understand ways in which technology can help explore the data. Learners should be able to demonstrate the ability to analyse a subset or features of the data using a calculator with standard statistical functions.