Nottingham High School


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		Further Maths

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Home : Senior School : Academic Departments : Maths : Schemes of Work : Further Maths

Further Maths

Pure Further Mathematics Syllabus 2003-2005

These notes are intended as a guide to the topics that feature under the new specifications.

Overall there is very little change from the previous system although certain topics have moved from Further Mathematics into the single subject specification.

The Text Book References column gives relevant page references from Backhouse and Houldsworth Pure Mathematics I and II, however some topics will need supplementing with exercises from elsewhere, see both the Pure Mathematics and the Further Pure Mathematics files in W13 for previously used supplementary sheets.

For more details on a particular topic see the Edexcel Specification booklet.

In the Notes column if a topic is bracketed, e.g. (Remainder Theorem), this means that this particular part of the specification will be covered elsewhere (either earlier or later).

Remember the emphasis on proof throughout the specifications. Students should be able to prove almost all of the major results, e.g. Binomial Theorem, De Moivre's Theorem, met during the course, as well as other results, e.g. AM-GM inequality, divisibility.

Specification Reference	Text Book References	Notes
P1 (2) Algebra	I p512-515; p173-179; p188-191; II p45-48	Indices, surds, Factor Theorem, quadratics, discriminant, identities, algebraic division, simultaneous equations, inequalities including modulus, symmetric functions of roots of quadratics. Do cover Remainder Theorem here.
P2 (5) Exponentials and logarithms	I p179-184; II p19-20; p26-29	Exponential and natural log functions, general exponentials, laws of logs, equations e.g. and .
P2 (1) Algebra and fractions; P3 (1) Algebra	II p48-p54; p87-96	Rational functions, partial fractions, (Remainder Theorem). Also, cover rational inequalities and curve sketching.
P1 (3) Trigonometry	I p363-374; p325-340	Radians, arc length, sector area, major (and minor) ratios for any angle, graphs, Pythagorean identities, simple identities and equations. Also, cover sine and cosine rules.
P2 (4) Trigonometry	I p377-379; p341-362; p382-385; II p220-227	(Minor ratios), inverse functions, compound, half and double angle formulae, factor formulae, form, equations, identities. Also, cover small angle approximations, general solutions and 3d questions.
P1 (1) Proof	I p266-268	Include induction.
P1 (5) Sequences and series	I p256-266; p271-274	Sequences, APs, GPs, S notation. Recurrence relations.
P2 (3) Sequences and series	I p275-283	(Recurrence relations), Binomial expansion for integer powers.
P3 (3) Series	I p283-287	Binomial expansion for rational powers.
P2 (2) Functions	I p22-p47	Domain, range, graphs, inverses, composition, modulus function, transformations of graphs.
P1 (4) Coordinate geometry in the (x,y) plane	I p1-21	Equations of lines, parallel and perpendicular lines, midpoints.
P1 (6) Differentiation	I p55 -73; p86-105; p124- 130	Differentiation of x^p where p is rational, second derivatives, increasing and decreasing functions, stationary points, curve sketching, tangents and normals. Also, cover Chain Rule here.
P2 (6) Differentiation	II p20-26; p29-34	Exponentials and logarithms. Include derivative of a^x.
P1 (7) Integration	I p106-123; p147-160	Antidifferentiation, integration of x^p where p is rational (), definite and indefinite integrals, areas bounded by axes.
P2 (7) Integration	II p23; p31-43; I p160-165	Exponentials, 1/x, volumes of revolution about coordinate axes.
P3 (2) Coordinate geometry in the (x,y) plane	I p405-414; p426-430	The circle, cartesian and parametric equations.
P3 (4) Differentiation	I p385-390; p130-139; p431-432; II p256-258	Trig functions, (Chain Rule), Product and Quotient Rules, inverse functions include trig, exponential growth and decay, implicit and parametric, formation of simple differential equations, connected rates of change.
P3 (5) Integration	I p387-392; II 1-18; p34-43; p54-58; p228-236; p238-241; p258-263	Trig functions, substitution, by parts, areas defined parametrically, use of partial fractions, use of trig identities, first order differential equations with separable variables. Include integration of inverse trig functions.
P3 (6) Vectors	I p288-313; p316-324	i, j, k vectors, position vectors, distance between points, vector equations of lines, scalar product, angles between lines.
P2 (8) Numerical methods	I p466-478; II p323-329; p335-337	Recurrence relations, sign change method, iterative solutions, Trapezium rule. Also, cover Simpson's rule.
P1, P2, P3 Specifications have now been completed.
P4 (1) Inequalities	All covered earlier.	(Rational inequalities, modulus inequalities.)
P4 (2) Series	I p268-271	Method of differences, standard series sums.
P6 (4) Maclaurin and Taylor series	II p309-322	Maclaurin and Taylor series, (series solutions of differential equations).
P4 (3) Complex numbers	I p200-211	Cartesian and polar forms, Argand diagram, roots of real quadratics, conjugate roots of real polynomials.
P5 (2) Hyperbolic functions; P5 (3) Differentiation; P5 (4) Integration	II p339-353	Definitions, graphs, identities, inverse functions, logarithmic equivalents, differentiation, integration, integration of inverse hyperbolics and trig functions, integration of quadratic surds. (Leave reduction formulae, arc length and surface area of revolution until later.)
P4 (5) First order differential equations	II p263-273	Formation, sketching family of solutions, integrating factor, substitution.
P4 (6) Second order differential equations	II p283-304	Auxiliary quadratic equation, complementary function, particular integral, substitution.
P4 (7) Polar coordinates	I p417-422; II p354-355	Sketching polar curves, tangents parallel to and perpendicular to initial line, area of sector.
P5 (1) Coordinate systems	I p393-404; p432-435; II p151-182; p364-369	Parabola, ellipse, hyperbola, rectangular hyperbola, focus-directrix properties, tangents and normals, loci, intrinsic coordinates, radius of curvature.
P5 (4) Integration (continued).	II p241-247; p356-364	Reduction formulae, arc length and surface area of revolution for cartesians and parametrics.
P4 (4) Numerical solutions of equations; P6 (5) Numerical methods	I p478-485; II p330-338	Interval bisection, linear interpolation, Newton-Raphson, step-by-step methods to solve differential equations. Also, cover Taylor series method for solving differential equations here.
P6 (1) Complex numbers	II p270-389	Euler's formula, relationship between trig functions and hyperbolics, De Moivre's Theorem, loci in the Argand diagram, transformations from the z-plane to the w-plane.
P6 (2) Matrix algebra	I p212-242; II p129-150	Linear transformations in 2 and 3 dimensions, transpose, 2 x 2 and 3 x 3 determinants and inverses, inverse transformations, eigenvalues and eigenvectors, diagonalising symmetric matrices.
P6 (3) Vectors	II p390-403	Vector product, triple scalar product, areas and volumes, points, lines and planes, distance from point to plane, line of intersection of planes, distance between skew lines. Also, cover the vector triple product.
P6 (6) Proof	Already covered throughout the course but should be revisited.	(Induction), divisibility, inequalities, general terms of sequences defined recursively, etc.

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