Use the equation of a plane in any of the forms ax + by + cz = d or r.n = p or r = a + λb + μc and convert equations of planes from one form to another as necessary in solving problems Recall that the vector product a × b of two vectors can be expressed either as absinθn, where n is a unit vector, or in component form ai+bj+ck Use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections, including – determining whether a line lies in a plane, is parallel to a plane or intersects a plane, and finding the point of intersection of a line and a plane when it exists – finding the foot of the perpendicular from a point to a plane – finding the angle between a line and a plane, and the angle between two planes – finding an equation for the line of intersection of two planes – calculating the shortest distance between two skew lines – finding an equation for the common perpendicular to two skew lines.

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

Empower your students and elevate your lessons with our expertly designed PPT and lesson booklet 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 for ∑r, ∑r^2 and ∑r^3 Use the standard series for for ∑r, ∑r^2 and ∑r^3 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.

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

The resource covers: Integration hyperbolic functions and inverses Derive and use reduction formulae for the evaluation of definite integrals 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 Use integration to find arc lengths for curves with equations in Cartesian coordinates, including the use of a parameter, or in polar coordinates 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.

The resource covers : Differentiating hyperbolic functions and inverses Differentiation implicit functions Differentiating parametric equations Differentiating inverse of trigonometric functions Maclaurin series

The resource covers: Integration hyperbolic functions and inverses Derive and use reduction formulae for the evaluation of definite integrals 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 Use integration to find arc lengths for curves with equations in Cartesian coordinates, including the use of a parameter, or in polar coordinates 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.

The resource covers : Differentiating hyperbolic functions and inverses Differentiation implicit functions Differentiating parametric equations Differentiating inverse of trigonometric functions Maclaurin series