Understand the concept of angular speed for a particle moving in a circle, and use the relation v = rw Understand that the acceleration of a particle moving in a circle with constant speed is directed towards the centre of the circle, and use the formulae r w^2 and v^2/r Solve problems which can be modelled by the motion of a particle moving in a horizontal circle with constant speed Solve problems which can be modelled by the motion of a particle in a vertical circle without loss of energy. Find a normal contact force or the tension in a string, locating points at which these are zero, and conditions for complete circular motion.

Model the motion of a projectile as a particle moving with constant acceleration and understand any limitations of the model Use horizontal and vertical equations of motion to solve problems on the motion of projectiles, including finding the magnitude and direction of the velocity at a given time or position, the range on a horizontal plane and the greatest height reached Derive and use the Cartesian equation of the trajectory of a projectile, including problems in which the initial speed and/or angle of projection may be unknown.

Use a probability density function which may be defined piecewise Use the general result E(g(x)) =∫f(x)g(x) dx where f(x) is the probability density function of the continuous random variable X and g(X) is a function of X Understand and use the relationship between the probability density function (PDF) and the cumulative distribution function (CDF), and use either to evaluate probabilities or percentiles Use cumulative distribution functions (CDFs) of related variables in simple cases e.g. given the CDF of a variable X, find the CDF of a related variable Y, and hence its PDF, e.g. where Y = X^ 3.

Sign Test PPT Paired Sign Test PPT One Sample Wilcoxon Sign Rank Test PPT Wilcoxon-Matched-Pairs Sign-Rank Test PPT Wilcoxon Rank-Sum Test PPT

Further Pure Maths 1 Further Maths 2 Further Mechanics Further Statistics

Use integration to find surface areas of revolution about one of the axes for curves with equations in Cartesian coordinates, including the use of a parameter.

Approximating area under a curve using area of rectangles and use rectangles to estimate or set bounds for the area under a curve or to derive inequalities or limits concerning sums

Derive and use reduction formulae for the evaluation of definite integrals

Integration hyperbolic functions and inverses

The resource covers Maclaurin’s series

The resource covers the differentiation of parametric equations

The resource covers differentiation of inverse trigonometric functions

The resource covers the differentiation of implicit functions

The resource covers the differentiating hyperbolic functions and inverses

Understand the concept of a probability generating function (PGF) and construct and use the PGF for given distributions e.g discrete uniform, binomial, geometric and Poisson distributions Use formulae for the mean and variance of a discrete random variable in terms of its PGF, and use these formulae to calculate the mean and variance of a given probability distribution Use the result that the PGF of the sum of independent random variables is the product of the PGFs of those random variables.

Determine a confidence interval for a population mean, based on a small sample from a normal population with unknown variance, using a t-distribution Determine a confidence interval for a difference of population means, using a t-distribution or a normal distribution, as appropriate.

Calculate a pooled estimate of a population variance from two samples Formulate hypotheses concerning the difference of population means, and apply, as appropriate – a 2-sample t-test – a test using a normal distribution

Formulate hypotheses and apply a hypothesis test concerning the population mean using a small sample drawn from a normal population of unknown variance, using a t-test

Formulate hypotheses concerning the difference of population means, and apply, as appropriate – a paired sample t-test – a test using a normal distribution

Use a single-sample sign test to test a hypothesis concerning a population median