Empower your students and elevate your lessons with our expertly designed lesson booklet/worksheet for teaching proof by induction. Cover these critical concepts with confidence: Summation of Series General Terms in Sequences Divisibility Rules Matrix Products Complex Numbers Reduction Formulae Finding the 𝑛𝑡ℎ Derivative

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.

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.

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 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 paired sample t-test – a test using a normal distribution 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.

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

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.

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.

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

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.

Fit a theoretical distribution, as prescribed by a given hypothesis, to given data Use a χ2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis Use a χ2-test, with the appropriate number of degrees of freedom, for independence in a contingency table.

Carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix Recall the meaning of the terms ‘singular’ and ‘non-singular’ as applied to square matrices and, for 2 x 2 and 3 x 3 matrices, evaluate determinants and find inverses of non-singular matrices understand and use the result, for non-singular matrices, (AB)^ –1 = B^ –1 A^-1 The notations det M for the determinant of a matrix M, and I for the identity matrix Understand the use of 2 x 2 matrices to represent certain geometric transformations in the x-y plane, in particular – understand the relationship between the transformations represented by A and A^–1 – recognise that the matrix product AB represents the transformation that results from the transformation represented by B followed by the transformation represented by A – recall how the area scale factor of a transformation is related to the determinant of the corresponding matrix – find the matrix that represents a given transformation or sequence of transformations Understand the meaning of ‘invariant’ as applied to points and lines in the context of transformations represented by matrices, and solve simple problems involving invariant points and invariant lines

The resource covers: Formulate a problem involving the solution of 3 linear simultaneous equations in 3 unknowns as a problem involving the solution of a matrix equation, or vice versa* Prove de Moivre’s theorem for a positive integer exponent Understand the cases that may arise concerning the consistency or inconsistency of 3 linear simultaneous equations, relate them to the singularity or otherwise of the corresponding matrix Solve consistent systems, and interpret geometrically in terms of lines and planes – to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle Understand the terms ‘characteristic equation’, ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices Find eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices express a square matrix in the form QDQ^–1, where D is a diagonal matrix of eigenvalues and Q is a matrix whose columns are eigenvectors, and use this expression Use the fact that a square matrix satisfies its own characteristic equation.

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

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

Sketch graphs of simple rational functions, including the determination of oblique asymptotes, in cases where the degree of the numerator and the denominator are at most 2 Show significant features of rational graphs, such as turning points, asymptotes and intersections with the axes. Determination of the set of values taken by the function, e.g. by the use of a discriminant. Understand and use relationships between the graphs of y = f(x), y^2 = f(x), y = 1/f(x) , y = If(x)I and y = f(IxI)