The resource covers: Understand de Moivre’s theorem, for a positive or negative integer exponent, in terms of the geometrical effect of multiplication and division of complex numbers Prove de Moivre’s theorem for a positive integer exponent Use de Moivre’s theorem for a positive or negative rational exponent – to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle – to express powers of sinθand cos θ in terms of multiple angles – in the summation of series – in finding and using the nth roots of unity

The resource covers: Find an integrating factor for a first order linear differential equation, and use an integrating factor to find the general solution Recall the meaning of the terms ‘complementary function’ and ‘particular integral’ in the context of linear differential equations, and recall that the general solution is the sum of the complementary function and a particular integral Find the complementary function for a first or second order linear differential equation with constant coefficients Recall the form of, and find, a particular integral for a first or second order linear differential equation in the cases where a polynomial or ae^bx or a cos px + b sin px is a suitable form, and in other simple cases find the appropriate coefficient(s) given a suitable form of particular integral. Use a given substitution to reduce a differential equation to a first or second order linear equation with constant coefficients or to a first order equation with separable variables. Use initial conditions to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation

Derive standard results ∑r, ∑r^2 and ∑r^3 Use the standard series to find related sums Use method of difference to finite sum of series Use partial fraction to find sum of series Find sum of infinity to convergent series

Derive standard results ∑r, ∑r^2 and ∑r^3 Use the standard series to find related sums Use method of difference to finite sum of series Use partial fraction to find sum of series Find sum of infinity to convergent series

Recall and use the relations between the roots and coefficients of polynomial equations Solve problems involving unknown coefficients in equations; restricted to equations of degree 2, 3 or 4 Use a substitution method to obtain an equation whose roots are related in a simple way to those of the original equation e.g where the new roots are reciprocals or squares or a simple linear function of the old roots.

Recall and use the relations between the roots and coefficients of polynomial equations Solve problems involving unknown coefficients in equations; restricted to equations of degree 2, 3 or 4 Use a substitution method to obtain an equation whose roots are related in a simple way to those of the original equation e.g where the new roots are reciprocals or squares or a simple linear function of the old roots.

Recall and use the relations between the roots and coefficients of polynomial equations Solve problems involving unknown coefficients in equations; restricted to equations of degree 2, 3 or 4 Use a substitution method to obtain an equation whose roots are related in a simple way to those of the original equation e.g where the new roots are reciprocals or squares or a simple linear function of the old roots.

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.

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.

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.

Calculate the moment of a force about a point Use the result that the effect of gravity on a rigid body is equivalent to a single force acting at the centre of mass of the body, and identify the position of the centre of mass of a uniform body using considerations of symmetry Use given information about the position of the centre of mass of a triangular lamina and other simple shapes Determine the position of the centre of mass of a composite body by considering an equivalent system of particles Use the principle that if a rigid body is in equilibrium under the action of coplanar forces then the vector sum of the forces is zero and the sum of the moments of the forces about any point is zero, and the converse of this Solve problems involving the equilibrium of a single rigid body under the action of coplanar forces, including those involving toppling or sliding.

Calculate the moment of a force about a point Use the result that the effect of gravity on a rigid body is equivalent to a single force acting at the centre of mass of the body, and identify the position of the centre of mass of a uniform body using considerations of symmetry Use given information about the position of the centre of mass of a triangular lamina and other simple shapes Determine the position of the centre of mass of a composite body by considering an equivalent system of particles Use the principle that if a rigid body is in equilibrium under the action of coplanar forces then the vector sum of the forces is zero and the sum of the moments of the forces about any point is zero, and the converse of this Solve problems involving the equilibrium of a single rigid body under the action of coplanar forces, including those involving toppling or sliding.

Calculate the moment of a force about a point Use the result that the effect of gravity on a rigid body is equivalent to a single force acting at the centre of mass of the body, and identify the position of the centre of mass of a uniform body using considerations of symmetry Use given information about the position of the centre of mass of a triangular lamina and other simple shapes Determine the position of the centre of mass of a composite body by considering an equivalent system of particles Use the principle that if a rigid body is in equilibrium under the action of coplanar forces then the vector sum of the forces is zero and the sum of the moments of the forces about any point is zero, and the converse of this Solve problems involving the equilibrium of a single rigid body under the action of coplanar forces, including those involving toppling or sliding.

Recall Newton’s experimental law and the definition of the coefficient of restitution, the property 0 ≤ e ≤ 1, and the meaning of the terms ‘perfectly elastic’ (e = 1) and ‘inelastic’ (e = 0) Use conservation of linear momentum and/or Newton’s experimental law to solve problems that may be modelled as the direct or oblique impact of two smooth spheres, or the direct or oblique impact of a smooth sphere with a fixed surface.

Recall Newton’s experimental law and the definition of the coefficient of restitution, the property 0 ≤ e ≤ 1, and the meaning of the terms ‘perfectly elastic’ (e = 1) and ‘inelastic’ (e = 0) Use conservation of linear momentum and/or Newton’s experimental law to solve problems that may be modelled as the direct or oblique impact of two smooth spheres, or the direct or oblique impact of a smooth sphere with a fixed surface.

Use Hooke’s law as a model relating the force in an elastic string or spring to the extension or compression, and understand the term modulus of elasticity Use the formula for the elastic potential energy stored in a string or spring Solve problems involving forces due to elastic strings or springs, including those where considerations of work and energy are needed

Recall Newton’s experimental law and the definition of the coefficient of restitution, the property 0 ≤ e ≤ 1, and the meaning of the terms ‘perfectly elastic’ (e = 1) and ‘inelastic’ (e = 0) Use conservation of linear momentum and/or Newton’s experimental law to solve problems that may be modelled as the direct or oblique impact of two smooth spheres, or the direct or oblique impact of a smooth sphere with a fixed surface.

Use Hooke’s law as a model relating the force in an elastic string or spring to the extension or compression, and understand the term modulus of elasticity Use the formula for the elastic potential energy stored in a string or spring Solve problems involving forces due to elastic strings or springs, including those where considerations of work and energy are needed

Use Hooke’s law as a model relating the force in an elastic string or spring to the extension or compression, and understand the term modulus of elasticity Use the formula for the elastic potential energy stored in a string or spring Solve problems involving forces due to elastic strings or springs, including those where considerations of work and energy are needed

Solve problems which can be modelled as the linear motion of a particle under the action of a variable force. Setting up and solving an appropriate differential equation involving variable force.