FP1 Study Guide

(2008 Specification)

COURSE OUTLINE

FP1 is a natural progression from C1/C2/C3/C4, some topics will be familiar to you and they will be looked at in more depth, for example Differential Equations. You will also meet some new topics such as Complex Numbers.

You will be expected to apply more than one topic as FP1 questions link various areas together and facts from C1/C2/C3/C4 are assumed knowledge throughout the syllabus.

Although you may initially find some topics rather abstract, all pure mathematics does have relevant applications in other fields. You may use this knowledge in the applied (Mechanics and Statistics) Units, or even beyond school mathematics. e.g. Electronics, engineering and psychology.

THE WORKThe unit will be covered in 3 hours of class time and about 6 hours of study time every week. Homework may consist of written exercises from the textbook, research, informal presentations, worksheets or past papers. All homework must be completed on time and submitted when required.

SPECIFICATION CONTENTS FOR FP1

TERM CONTENTAUTUMN Series

Complex Numbers

SPRING Numerical Solutions Coordinate Systems (Conic Sections)

SUMMER Matrix Algebra Proof

Revision Papers

You will be tested on the content covered each half term and be awarded an internal grade to assess progress.

Click here for a FP1 Mindmap

Notes:-

SCHEME OF WORK FOR AS - FURTHER MATHS MODULE FP1

Text : - Heinemann ‘FP1 for Edexcel’ (OLD FP1) & Further Pure Mathematics 1 (Pearson)

ISBN-10: 0435511092 ISBN-978-0-435519-23-0

ISBN-13: 978 0435511098

There are also linked Exercised from PURE MATHEMATICS 4 Syllabus 1992-04Authors G Mannall and M Kenwood

CHAPT. TOPIC & ASSESSMENT (New Texts)

TIME(wks)

OBJECTIVES (FP1 Specification)

SPECIFICATION NOTES METHODOLOGY OTHER RESOURCES

ASSESSMENT(Old Texts)

Ch. 2P. 11-19(OLD FP1)

Ch. 5P.107-121(NEW FP1)

1. Series(NEW FP1 Text)

Ex. 5A, pg 109 (Sigma Notation Revision)

Ex 5B, pg 111Ex 5C pg 113(Using standard result for

only

Ex 5D-F, pg 115-121Using Standard Results for

2

Summation of simple finite series.

Standard results may be quoted from the Formula Book.

Method of Differences is not required

Candidates should be able to sum series such as

Proof by induction is not required yet (See topic 6.Proof)

N.B. Σ 1 (from 1 to n) = n

Review C1/C2 use of the Σ symbol for A.P & G.P.Show that Σr is really an A.P with a=1, d=1. Run through C1 proof to illustrate the formula in the book.

Review Partial Fractions (C4) to break down complicated ‘r’ fractions to be summed.

Summing Series PowerPoint

(Old FP1)Ex 2A pg 15 (Using Identities and Numerical Substitution)Ex 2B pg 18 Using Standard Results for

CHAPT. TOPIC & ASSESSMENT

(New Texts)

TIME(wks)


SPECIFICATION NOTES METHODOLOGY OTHER RESOURCE

ASSESSMENT

Ch. 3P. 21-48(OLD FP1)

Ch. 1P. 1- 31(NEW FP1)

2. Complex Numbers(NEW FP1 Text)

Ex 1A-C. pg 4-10(Manipulating Complex Numbers and Solving “Complex” Quadratics)

Ex 1D, pg 14(Argand Diagram)

E 1E & 1F, pg 18-21(Modulus – Argument)

Ex 1G & 1Hpg 23-30(Harder Equations and Conjugate Roots)

3Definition of complex numbers in the form a + ib

and r cos + i r sin .

Sum, product and quotient of complex numbers.

Complex solutions of quadratic equations with real coefficients.

Conjugate complex roots of polynomial equations with real coefficients.

The meaning of conjugate, modulus, argument, real part, imaginary part and equality of complex numbers should be known.

=

Geometrical representation of complex numbers in the Argand diagram.

Geometrical representation of sums, products and quotients of complex numbers.Link with vectors covered in C4.

Knowledge that if z1 is a root of f(z) = 0 then z1

* is also a root.

Link with Polar

Begin by looking at a previously ‘unsolvable’ Quadratic, i.e. b2 – 4ac < 0.Sketch the graph and show that it does not cross the x-axis (no real roots). Put across the idea that we need an imaginary axis to cross for some sort of root to exist.

Define i as √(-1). Hence i2 =1.Link Argand Diagrams with i, j vectors (C4/M1) and Modulus/Arg with Magnitude and Direction.

Use difference of two squares and Surds (Rationalisation) to illustrate the method of manipulation of Complex Conjugates.

(See ‘Complex Numbers’ section of the Open University Maths DVD.)

Complex Numbers - The Basics

Complex Numbers - Loci examples using Argand diagrams

(Old FP1)Ex 3A pg 26 (Manipulating Complex Numbers and Solving “Complex” Quadratics)

(Old FP1)Ex 3B pg 34 (Argand Diagrams)

Ex 3C pg 41 (Modulus – Argument)

Ex 3D pg 46 (Harder Equations and Conjugate Roots)

Coordinates (FP2)

CHAPT. TOPIC & ASSESSMENT

(New Texts)

TIME(wks)



ASSESSMENT

Ch. 4P. 51-65(OLD FP1)

Ch. 2P. 32-40(NEW FP1)

3. Numerical Solutions of Equations(NEW Text)

Ex 2A, pg. 34(Int. Bisection)

Ex 2B, pg. 37(Lin. Interpolation)

Ex 2C, pg. 39 (Newton–Raphson)

Ex 2D, pg. 40(Mixed Questions)

1-2Equations of the form f(x) = 0 solved numerically by:-(i) interval bisection,

(ii) linear interpolation

(iii) Newton-Raphsonprocess.

Begins with a review of Iterations from C3.

This chapter is about functions which will not iterate.

Therefore we need other methods to find the roots.

Show that the roots are where the graph crosses the x-axis.

Use Omnigraph to zoom in to the solutions as the interval bisects.

Construct tangents using Omnigraph to illustrate how Newton-Raphson works

Interval Bisection

Interval Bisection and Linear Interpolation

Newton-Raphson

(Old FP1) Ex 4A pg 61 (Mixed Exercise with all 3 methods)

Ch. 3P. 41-62(NEW PP1

4. Coordinate Systems (Conic Sections)

2-3Cartesian equations for the Parabola and Rectangular Hyperbola.

Idea of parametric equation for parabola and rectangular hyperbola.

The focus-directrix property of the parabola.

Tangents and normals to these curves.

The idea of (at2, 2at) as a general point on the parabola is all that is required.

Concept of focus and directrix and parabola as locus of points equidistant from focus and directrix.

Differentiation of

y = , y = .

Revise Parametric Equations from C4.

Candidates should be familiar with the equations:

y2 = 4ax or x = at2, y = 2at (Parabola)

and

or .

(Hyperbola)

Use Omnigraph and

Some notes and diagrams about Conic Sections.

Sketchpad illustration of the Parabola

(NEW FP1 Text)Ex 3A, pg44-45 (Revision of C4 Parametric Equations)

Ex 3B, pg 48(General Parabola Equation)

Ex 3C, pg50-51 (Parabola Problems involving Tangents etc.)

Ex 3D, pg 56(Hyperbola)

Ex 3E, pg 59-60

See FP3 for the Ellipse.

Parametric differentiation is not required.

Geometers Sketchpad to illustrate Focus-Directrix properties.

(Parabola & Hyperbola Problems)

CHAPT. TOPIC TIME(wks)



ASSESSMENT

Ch. 4P. 72-106(NEW FP1)

5. Matrix Algebra

2-3Addition and subtraction of matrices.

Multiplication of a matrix by a scalar.Products of matrices.Evaluation of 2 × 2 determinants.

Inverse of 2 × 2 matrices.

Linear transformations of column vectors in two dimensions and their matrix representation.

Define Singular and non-singular matrices.Use of the relation

Use the Row x Column Rule for Multiplication (Hence RC – Roman Catholic!)

(AB)–1 = B–1A–1.

Using Matrices to solve Simultaneous Equations.

Applications of matrices to geometrical transformations.

Stress the importance of thinking in terms of Rows and Columns.

Show the Identity (2 x 2) matrix.

Inverse Matrices and their effects on Transformations

Identification and use of the matrix representation of the following single and combined transformations.

a) Reflection in coordinate axes and lines y = +x,

b) Rotation of multiples of 45 about (0, 0).

c) Enlargement about centre (0, 0), with scale factor, (k ≠ 0), where

Matrix Addition & Multiplication

Summary of different 2 x 2 Transformation Matrices

Flash Tool to show all the matrix transformations

(NEW FP1 Text)

Ex 4A & 4B pg75-77 (Addition of Matrices)

Ex 4C, pg 81-82(Multiplication of Matrices)

Ex 4G, pg 97-98 (Inverse of a Matrix)

Ex 4D-4F, pg85-94 (Transformations using Matrices)

Ex 4H, pg100-101(Inverse Transformation Matrices)

E 4I, pg 102-103 (Areas by Determinants)

Ex 4J, pg 102-103

k ℝ. (Simultaneous Equations)

CHAPT. TOPIC TIME(wks)



ASSESSMENT

Ch. 6P.122-136(NEW FP1)

6. Proof (By Induction)

2Proof by the Method of Induction. (Four types of examples)

* Summation of series

e.g. Show

or

* Matrix Products

e.g. show

To include induction proofs for

* Divisibility e.g. show is

divisible by 4.

* Finding general terms in a sequencee.g. if

with , prove that

.

Revise the followingChapters:- Series & Matrices (FP1), and Iterations (C3)

Show that all the situations basically follow the same procedure.

1) Assume the statement to prove is true for n = k (where k is a positive integer)

2) Write an expression for the next term i.e. n = k + 1 (by writing the kth term and adding the next term)

3) Manipulate this expression and simplify it to look like the original expression with a (k+1) replacing all the k’s.

4) This has proved that the situation works for k, k+1 etc.

5) Therefore it can be induced that it would work for k+1 & k+2 in a similar way. Hence it works for all positive integers k.

Mathematical Induction Notes & Examples

Mathematical Induction PowerPoint

(NEW FP1 Text)

Ex 6A, pg 127(Proof by Induction for Sigma types)

Ex 6B, pg. 130(Divisibility Proofs)

Ex 6C, pg 132 (Recurrence Relations)

Ex 6D, pg 134 (Matrix Multiplication)

Ex 6E, pg 135 (Mixed Questions)

THE EXAMINATION

GRADE BOUNDARIES

Grade boundaries vary according to the unit and the year, but a rough guide is:

Required Marks Grade 80% A 70% B 60% C 50% D 40% E 30% N

20% U

(All internal tests will be assessed using this criteria)

THE EXAM PAPER

The paper has 75 available marks and is 90 minutes long – so it’s just over one minute per mark.There are usually 8-10 questions on the exam paper and you must answer every question, there is no choice in the questions. Like GCSE exam, you write on the question paper. There is a formula book provided for the examination. Make sure you are familiar with its contents before the test. It contains the formulae for all previous Units

NOTE :- New regulations will allow each unit to be retaken (more than once) with the better result counting. Candidates will receive a statement of results for each unit. A candidate who has sat the three AS individual units may also resit the whole qualification.

To be awarded AS-Further Mathematics you need to take FP1 plus two additional ‘applications’ Units .

RESOURCES

BOOKS - SCHOOL LIBRARY

Advanced level mathematics. Solomon R.C.Pure and applied mathematics 1. Dawson A.Pure mathematics. Bostock L.Understanding Pure Mathematics Sadler A.J.Essential Mathematics for A-Level Taylor D.C.Essential Pure Mathematics Backhouse J.K.Four Figure Tables Godfrey CComplete O’Level Mathematics GreerMathematics Encyclopaedia Smeltzer PModular Mathematics BostockAdvanced Mathematics 1 Celia C.W.

MATHEMATICS DEPARTMENT

Modular Mathematics (A) Pure : Bostock & ChandlerModular Mathematics (B) Pure : Bostock & ChandlerThe Core Course for A/L : Bostock & Chandler

Heinemann Modular Mathematics for London : P1Heinemann Modular Mathematics for London : P2Heinemann Modular Mathematics for London : P3Heinemann Modular Mathematics for London : P4Mathematical Formulae Books

Revision Folders P1, P2 Box Files for P1 and P2 : National Extension College

Formulae given in the Formula Booklet

RECORD OF PROGRESS FOR THE FP1 COURSE

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ICT RESOURCES

This Study Guide contains hyperlinks (in blue font within the Scheme of Work section) which take you to helpful PowerPoints showing further examples.

Past Examination Papers published by EDEXCEL are also available via the SUBJECTS folder on the desktop. Click on SUBJECTS>>>MATHEMATICS>>>WORKSHEETS>>>PAST A-LEVEL PAPERS. Then open the folder for the course you are following. These papers are in Word or Acrobat Reader Format and should either be saved in your own home h-drive or e-mailed to your own PC at home as an attached document via say, hotmail. Alternatively, you can print them off and work on paper!

Although some of your lessons will be booked in one of the ICT rooms (or interactive whiteboards) for lessons involving graphical work, Omnigraph is an extremely powerful tool you can access freely at any time. It can be found via SUBJECTS>>>MATHEMATICS>>>APPLICATIONS>>> OMNIGRAPH.

Try the following website www.mathsnet.net Follow the links to AS/A2 and you will find a whole host of resources, notes and on-line papers.

Finally make use of the EDEXCEL website (www.edexcel.org.uk) It has useful information about YOUR mathematics course.

http://www.edexcel.org.uk/

http://www.mathsnet.net/

FP1 Study Guide

Documents

Transcript of FP1 Study Guide