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1. Further Pure Mathematics 1

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  1. 1.1 Roots of polynomial equations

    1. 1.1.1Polynomial roots

      • recall and use the relations between the roots and coefficients of polynomial equations e.g. to evaluate symmetric functions of the roots or to solve problems involving unknown coefficients in equations; restricted to equations of degree 2, 3 or 4 only.

    2. 1.1.2Root transformations

      • use a substitution to obtain an equation whose roots are related in a simple way to those of the original equation Substitutions will not be given for the easiest cases, e.g. where the new roots are reciprocals or squares or a simple linear function of the old roots.

  2. 1.2 Rational functions and graphs

    1. 1.2.1Rational graphs

      • 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 Including determination of the set of values taken by the function, e.g. by the use of a discriminant. Detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as turning points, asymptotes and intersections with the axes.

    2. 1.2.2Related graphs

      • understand and use relationships between the graphs of y = f(x), y2 = f(x), y x 1 f= ^h, y xf= ^h and y xf= ^h. Including use of such sketch graphs in the course of solving equations or inequalities.

  3. 1.3 Summation of series

    1. 1.3.1Standard series sums

      • use the standard results for r/, r2/, r3/ to find related sums

    2. 1.3.2Method of differences

      • use the method of differences to obtain the sum of a finite series Use of partial fractions to express a general term in a suitable form may be required.

    3. 1.3.3Series convergence

      • recognise, by direct consideration of a sum to n terms, when a series is convergent, and find the sum to infinity in such cases

  4. 1.4 Matrices

    1. 1.4.1Matrix operations

      • carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix Including non-square matrices. Matrices will have at most 3 rows and columns.

    2. 1.4.2Determinants and inverses

      • recall the meaning of the terms 'singular' and 'non-singular' as applied to square matrices and, for 22# and 33# matrices, evaluate determinants and find inverses of non-singular matrices The notations det M for the determinant of a matrix M, and I for the identity matrix, will be used.

    3. 1.4.3Inverse matrix products

      • understand and use the result, for non-singular matrices, (AB)-1 = B-1A-1 Extension to the product of more than two matrices may be required.

    4. 1.4.4Matrix transformations

      • understand the use of 22# 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 Understanding of the terms 'rotation', 'reflection', 'enlargement', 'stretch' and 'shear' for 2D transformations will be required. Other 2D transformations may be included, but no particular knowledge of them is expected.

    5. 1.4.5Invariant points and lines

      • 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. e.g. to locate the invariant points of the transformation represented by 6 2 5 3eo, or to find the invariant lines through the origin for 4 2 1 1 - eo, or to show that any line with gradient 1 is invariant for 2 1 0 1eo.

  5. 1.5 Polar coordinates

    1. 1.5.1Polar coordinates

      • understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa The convention r 0H will be used.

    2. 1.5.2Polar curves

      • sketch simple polar curves, for 02 <G ri or 1 G-rr i or a subset of either of these intervals Detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as symmetry, coordinates of intersections with the initial line, the form of the curve at the pole and least/greatest values of r.

    3. 1.5.3Polar area

      • recall the formula r d2 1 2 iy for the area of a sector, and use this formula in simple cases.

  6. 1.6 Vectors

    1. 1.6.1Plane equations

      • 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

    2. 1.6.2Vector product

      • recall that the vector product ab# of two vectors can be expressed either as sinab ni t, where nt is a unit vector, or in component form as ab ab ab abij23 32 31 13-- +`` jj ab ab k12 21-+`j

    3. 1.6.3Lines and planes

      • 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.

  7. 1.7 Proof by induction

    1. 1.7.1Proof by induction

      • use the method of mathematical induction to establish a given result e.g. rn n4 1 1 r n 42 2 1 = + = ^h/, u 2 1 13n n 1= + -_ i for the sequence given by uu 31nn1 -=+ and u 11 =, 4 6 1 1 32 2 32 6 12 32 n n n n n11 # # - - - - - -= ++f fp p, 32 53nn2 # -+ is divisible by 8.

    2. 1.7.2Inductive conjecture

      • recognise situations where conjecture based on a limited trial followed by inductive proof is a useful strategy, and carry this out in simple cases. e.g. find the nth derivative of x ex, find !rr r n 1 # = /.