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

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 Wilcoxon matched-pairs signed-rank test as appropriate, to test for identity of populations.

Use a a single-sample Wilcoxon signed-rank test to test a hypothesis concerning a population median

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

Use a Wilcoxon rank-sum test, as appropriate, to test for identity of populations.

Use a paired-sample sign test as appropriate, to test for identity of populations.

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)

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)

Understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa Sketch simple polar curves, for 0≤θ<2π or -π≤θ<π or a subset of either of these interval Recall the formula 1/2 ∫r^2 dθ for the area of a sector and use this formula in simple cases.

Understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa Sketch simple polar curves, for 0≤θ<2π or -π≤θ<π or a subset of either of these interval Recall the formula 1/2 ∫r^2 dθ for the area of a sector and use this formula in simple cases.

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.

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

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

Understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa Sketch simple polar curves, for 0≤θ<2π or -π≤θ<π or a subset of either of these interval Recall the formula 1/2 ∫r^2 dθ for the area of a sector and use this formula in simple cases.

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

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: 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.