<ul> <li>Carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix</li> <li>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</li> <li>The notations det M for the determinant of a matrix M, and I for the identity matrix</li> <li>Understand the use of 2 x 2 matrices to represent certain geometric transformations in the x-y plane, in particular<br /> – understand the relationship between the transformations represented by A and A^–1<br /> – recognise that the matrix product AB represents the transformation that results from the transformation represented by B followed by the transformation represented by A<br /> – recall how the area scale factor of a transformation is related to the determinant of the corresponding matrix<br /> – find the matrix that represents a given transformation or sequence of transformations</li> <li>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</li> </ul>

<ul> <li>Carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix</li> <li>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</li> <li>The notations det M for the determinant of a matrix M, and I for the identity matrix</li> <li>Understand the use of 2 x 2 matrices to represent certain geometric transformations in the x-y plane, in particular<br /> – understand the relationship between the transformations represented by A and A^–1<br /> – recognise that the matrix product AB represents the transformation that results from the transformation represented by B followed by the transformation represented by A<br /> – recall how the area scale factor of a transformation is related to the determinant of the corresponding matrix<br /> – find the matrix that represents a given transformation or sequence of transformations</li> <li>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</li> </ul>

<ul> <li>Use the equation of a plane in any of the forms ax + by + cz = d or r.n = p<br /> or r = a + λb + μc and convert equations of planes from one form to another as necessary in solving problems</li> <li>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</li> <li>Use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections, including<br /> – 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<br /> – finding the foot of the perpendicular from a point to a plane<br /> – finding the angle between a line and a plane, and the angle between two planes<br /> – finding an equation for the line of intersection of two planes<br /> – calculating the shortest distance between two skew lines<br /> – finding an equation for the common perpendicular to two skew lines.</li> </ul>

<ul> <li>Carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix</li> <li>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</li> <li>The notations det M for the determinant of a matrix M, and I for the identity matrix</li> <li>Understand the use of 2 x 2 matrices to represent certain geometric transformations in the x-y plane, in particular<br /> – understand the relationship between the transformations represented by A and A^–1<br /> – recognise that the matrix product AB represents the transformation that results from the transformation represented by B followed by the transformation represented by A<br /> – recall how the area scale factor of a transformation is related to the determinant of the corresponding matrix<br /> – find the matrix that represents a given transformation or sequence of transformations</li> <li>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</li> </ul>

<p>The resource covers:</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>Understand the terms ‘characteristic equation’, ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices</p> </li> <li> <p>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</p> </li> <li> <p>Use the fact that a square matrix satisfies its own characteristic equation.</p> </li> </ul>

<p>The resource covers:</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>Understand the terms ‘characteristic equation’, ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices</p> </li> <li> <p>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</p> </li> <li> <p>Use the fact that a square matrix satisfies its own characteristic equation.</p> </li> </ul>

<p>The resource covers:</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>Understand the terms ‘characteristic equation’, ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices</p> </li> <li> <p>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</p> </li> <li> <p>Use the fact that a square matrix satisfies its own characteristic equation.</p> </li> </ul>

<p>The resource covers:</p> <ul> <li>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</li> <li>Prove de Moivre’s theorem for a positive integer exponent</li> <li>Use de Moivre’s theorem for a positive or negative rational exponent<br /> – to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle<br /> – to express powers of sinθand cos θ in terms of multiple angles<br /> – in the summation of series<br /> – in finding and using the nth roots of unity</li> </ul>

<p>The resource covers:</p> <ul> <li>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</li> <li>Prove de Moivre’s theorem for a positive integer exponent</li> <li>Use de Moivre’s theorem for a positive or negative rational exponent<br /> – to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle<br /> – to express powers of sinθand cos θ in terms of multiple angles<br /> – in the summation of series<br /> – in finding and using the nth roots of unity</li> </ul>

<p>The resource covers:</p> <ul> <li> <p>Find an integrating factor for a first order linear differential equation, and use an integrating factor to find the general solution</p> </li> <li> <p>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</p> </li> <li> <p>Find the complementary function for a first or second order linear differential equation with constant coefficients</p> </li> <li> <p>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.</p> </li> <li> <p>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.</p> </li> <li> <p>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</p> </li> </ul>

<p>The resource covers:</p> <ul> <li>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</li> <li>Prove de Moivre’s theorem for a positive integer exponent</li> <li>Use de Moivre’s theorem for a positive or negative rational exponent<br /> – to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle<br /> – to express powers of sinθand cos θ in terms of multiple angles<br /> – in the summation of series<br /> – in finding and using the nth roots of unity</li> </ul>

<p>The resource covers:</p> <ul> <li> <p>Find an integrating factor for a first order linear differential equation, and use an integrating factor to find the general solution</p> </li> <li> <p>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</p> </li> <li> <p>Find the complementary function for a first or second order linear differential equation with constant coefficients</p> </li> <li> <p>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.</p> </li> <li> <p>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.</p> </li> <li> <p>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</p> </li> </ul>

<p>The resource covers:</p> <ul> <li> <p>Find an integrating factor for a first order linear differential equation, and use an integrating factor to find the general solution</p> </li> <li> <p>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</p> </li> <li> <p>Find the complementary function for a first or second order linear differential equation with constant coefficients</p> </li> <li> <p>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.</p> </li> <li> <p>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.</p> </li> <li> <p>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</p> </li> </ul>

<p>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.</p>

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

<p>Derive and use reduction formulae for the evaluation of definite integrals</p>

<p>Integration hyperbolic functions and inverses</p>

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

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

<p>The resource covers:</p> <ul> <li>Integration hyperbolic functions and inverses</li> <li>Derive and use reduction formulae for the evaluation of definite integrals</li> <li>Approximating area under a curve using area of rectangles and use rectangles to estimate or set bounds for the area<br /> under a curve or to derive inequalities or limits concerning sums</li> <li>Use integration to find arc lengths for curves with equations in Cartesian coordinates, including the use of a<br /> parameter, or in polar coordinates</li> <li>Use integration to find surface areas of revolution about one of the axes for curves with equations in Cartesian<br /> coordinates, including the use of a parameter.</li> </ul>

<p>The resource covers Maclaurin’s series</p>