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12 Pure Core 1

Introduction

Candidates will be required to demonstrate:

a. construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’ and notation such as ∴ , ⇒, ⇐ and ⇔ .

Candidates are not allowed to use a calculator in the assessment unit for this module.

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Quadratic Equations has roots
Circles A circle, centre (,) and radius , has equation
Differentiation
function derivative
is a whole number
Integration
function integral
is a whole number
Area Area under a curve =

12.1 Algebra

Use and manipulation of surds. To include simplification and rationalisation of the denominator of a fraction.

Eg

Quadratic functions and their graphs. To include reference to the vertex and line of symmetry of the graph.
The discriminant of a quadratic function. To include the conditions for equal roots, for distinct real roots and for no real roots
Factorisation of quadratic polynomials. Eg factorisation of
Completing the square. Eg
Solution of quadratic equations.

Use of any of factorisation, or
completing the sqaure will be accepted.

Simultaneous equations, e.g. one linear and one quadratic, analytical solution by substitution.
Solution of linear and quadratic inequalities. Eg
Algebraic manipulation of polynomials, including expanding brackets and collecting like terms.
Simple algebraic division. Applied to a quadratic or a cubic polynomial divided by a linear term of the form where is a small whole number. Any method will be accepted, e.g. by inspection, by equating coefficients or by formal division e.g.
Use of the Remainder Theorem. Knowledge that when a quadratic or cubic polynomial is divided
by the remainder is and, that when , then is a factor and vice versa.
Use of the Factor Theorem. Greatest level of difficulty as indicated by , i.e. a
cubic always with a factor where is a small whole number but including the cases of three distinct linear factors, repeated linear factors or a quadratic factor which cannot be factorized in the real numbers.
Graphs of functions; sketching curves defined by simple equations. Linear, quadratic and cubic functions. The notation may be used but only a very general idea of the concept of a function is required. Domain and range are not included. Graphs of circles are included.
Geometrical interpretation of algebraic solution of equations and use of intersection points of graphs of functions to solve equations. Interpreting the solutions of equations as the intersection points of graphs and vice versa.
Knowledge of the effect of translations on graphs and their equations. Applied to quadratic graphs and circles, i.e. as a translation of and as a translation
of .

12.2 Coordinate Geometry

Equation of a straight line, including the forms and . To include problems using gradients, mid-points and the distance
between two points. The form is also included.
Conditions for two straight lines to be parallel or perpendicular to each other. Knowledge that the product of the gradients of two perpendicular lines is .
Coordinate geometry of the circle. Candidates will be expected to complete the square to find the centre and radius of a circle where the equation of the circle is for example given as .
The equation of a circle in the form The use of the following circle properties is required:
(i) the angle in a semicircle is a right angle;
(ii) the perpendicular from the centre to a chord bisects the chord;
(iii) the tangent to a circle is perpendicular to the radius at its point of contact.
The equation of the tangent and normal at a given point to a circle. Implicit differentiation is not required. Candidates will be expected to use the coordinates of the centre and a point on the circle or of other appropriate points to find relevant gradients.
The intersection of a straight line and a curve. Using algebraic methods. Candidates will be expected to interpret the geometrical implication of equal roots, distinct real roots or no real roots. Applications will be to either circles or graphs of quadratic functions.

12.3 Differentiation

The derivative of as the gradient of the tangent to the graph of at a point; the gradient of the tangent as a limit; interpretation as a rate of change. The notations or will be used.
A general appreciation only of the derivative when interpreting it is required. Differentiation from first principles will not be tested.
Differentiation of polynomials.
Applications of differentiation to gradients, tangents and normals, maxima and minima and stationary points, increasing and decreasing functions. Questions will not be set requiring the determination of or knowledge of points of inflection. Questions may be set in the form of a practical problem where a function of a single variable has to be optimised.
Second order derivatives. Application to determining maxima and minima.

12.4 Integration

Indefinite integration as the reverse of differentiation Integration of polynomials.
Evaluation of definite integrals. Interpretation of the definite integral as the area under a curve. Integration to determine the area of a region between a curve and the -axis. To include regions wholly below the -axis, i.e. knowledge that the integral will give a negative value.
Questions involving regions partially above and below the -axis will not be set. Questions may involve finding the area of a region bounded by a straight line and a curve, or by two curves.

13 Pure Core 2

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the module Core 1. Candidates will be required to demonstrate:

a. Construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’ and notation such as ∴ , ⇒, ⇐ and ⇔ .

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Trigonometry In the triangle

Laws of Logarithms

Differentiation
Function derivative
,      is a rational number
Integration
Function integral
,      is a rational number,

13.1 Algebra and Functions

Laws of indices for all rational exponents.
Knowledge of the effect of simple transformations on the graph of as represented by
.
Candidates are expected to use the terms reflection, translation and stretch in the direction in their descriptions of these transformations.

Eg graphs of ; ; ;

Descriptions involving combinations of more than one transformation will not be tested.

13.2 Sequences and Series

Sequences, including those given by a formula for the th term. To include notation for sums of series.
Sequences generated by a simple relation of the form To include their use in finding of a limit as by putting .
Arithmetic series, including the formula for the sum of the first natural numbers.
The sum of a finite geometric series.
The sum to infinity of a convergent geometric series. Candidates should be familiar with the notation in this context.
The binomial expansion of for positive integer . To include the notations and . Use of Pascal’s triangle or formulae to expand will be accepted.

13.3 Trigonometry

The sine and cosine rules.
The area of a triangle in the
form .
Degree and radian measure.
Arc length, area of a sector of a circle. Knowledge of the formulae .
Sine, cosine and tangent functions. Their graphs,symmetries and periodicity. The concepts of odd and even functions are not required.
Knowledge and use of

and .
Solution of simple trigonometric equations in a given interval of degrees or radians. Maximum level of difficulty as indicated by
and

13.4 Exponentials and logarithms

and its graph. Using the laws of indices where appropriate.
Logarithms and the laws of
logarithms.

The equivalence of and

The solution of equations of
the form .
Use of a calculator logarithm function to solve for example .

13.5 Differentiation

Differentiation of , where
is a rational number, and
related sums and differences.
i.e. expressions such as , including terms which can be

expressed as a single power such as .
Applications to techniques included in module Core 1.

13.6 Integration

Integration of , and related sums and differences. i.e. expressions such as or
Applications to techniques included in module Core 1.
Approximation of the area under a curve using the trapezium rule. The term 'ordinate' will be used. To include a graphical determination of whether the rule over- or under- estimates the area and improvement of an estimate by increasing the number of steps.

14 Pure Core 3

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1 and Core 2.

Candidates will be required to demonstrate:

a. construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’ and notation such as ∴ , ⇒, ⇐ and ⇔ ;

c. methods of proof, including proof by contradiction and disproof by counter-example.

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Trigonometry

Differentiation
function derivative












Volumes Volume of solid of revolution:

About the -axis:

About the -axis:

Integration
Function integral

14.1 Algebra and Functions

Definition of a function.

Domain and range of a function.

Notation such as may be used.

Domain may be expressed asfor example and range may be expressed as for example.

Composition of functions.
Inverse functions and their graphs. The notation will be used for the inverse of .
To include reflection in .
The modulus function. To include related graphs and the solution from them of inequalities such as using solutions of .
Combinations of the transformations on the graph of as represented by , , , . For example the transformations of: leading to ; leading to ; leading to

Transformations on the graphs of functions included in modules Core 1 and Core 2.

14.2 Trigonometry

Knowledge of
and functions.

Understanding of their domains and graphs.

Knowledge that

The graphs of these functions as reflections of the relevant parts of trigonometric graphs in are included. The addition formulae for inverse functions are not required.
Knowledge of secant, cosecant and cotangent. Their relationships to cosine, sine and tangent functions. Understanding of their domains and graphs.
Knowledge and use of , . Use in simple identities.

Solution of trigonometric equations in a given interval, using these identities.

14.3 Exponentials and Logarithms

The function and its graph.
The function and its graph; as the inverse function of .

14.4 Differentiation

Differentiation of
and
linear combinations of these functions.
Differentiation using the product rule, the quotient rule, the chain rule and by the use of .

E.g

Eg A curve has equation Find when

14.5 Integration

Integration of ,
Simple cases of integration:
by inspection or substitution;
Eg
by substitution; Eg
and integration by parts. Eg
These methods as the reverse processes of the chain and product rules respectively. Including the use of by inspection or substitution.
Evaluation of a volume of
revolution.
The axes of revolution will be restricted to the -axis and -axis.

14.6 Numerical Methods

Location of roots of by considering changes of sign of in an interval of in which is continuous.
Approximate solutions of equations using simple iterative methods, including recurrence relations of the form

Rearrangement of equations to the form
Staircase and cobweb diagrams to illustrate the iteration and their use in considerations of convergence.

Numerical integration of functions using the mid-ordinate rule and Simpson's rule. To include improvement of an estimate by increasing the number of steps.

15 Pure Core 4

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2 and Core 3.

Candidates will be required to demonstrate:

a: construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b: correct understanding and use of mathematical language and grammar in respect of terms such as 'equals', 'identically equals', 'therefore', 'because', 'implies', 'is implied by', 'necessary', 'sufficient' and notation such as ∴ , ⇒ , ⇐ and ⇔ ;

c: methods of proof, including proof by contradiction and disproof by counter-example.

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

 

Trigonometry

, where and

, where and

Vectors

15.1 Algebra and functions

Rational functions.

Including use of the Factor and Remainder Theorem for divisors of
the form .

Simplification of rational
expressions including
factorising and cancelling.

Expressions of the type

Algebraic division.

Any method will be accepted, e.g. by inspection, by equating coefficients or by formal division.

;

by using the given identity

Partial fractions
(denominators not more
complicated than repeated
linear terms).

Greatest level of difficulty

Irreducible quadratic factors will not be tested.

15.2 Coordinate geometry in the (x, y) plane

Cartesian and parametric equations of curves and conversion between the two forms.

Eg ;   , ;

15.3 Sequences and series

Binomial series for any
rational .

Expansion of

Greatest level of difficulty , expansion

valid for  

Series expansion of rational
functions including the use
of partial fractions

Greatest level of difficulty .

15.4 Trigonometry

Use of formulae for
and
and of expressions for
in the
equivalent forms of
or
.

Use in simple identities.

Solution of trigonometric equations in a given interval
Eg
Knowledge and use of double
angle formulae.
Knowledge that



is expected.

Use in simple identities.
For example,

Solution of trigonometric equations in a given interval. For example, solve

Use in integration. For example

15.5 Exponentials and Logarithms

Exponential growth and
decay
The use of exponential functions as models.

15.6 Differentiation and Integration

Formation of simple differential equations.To include the context of growth and decay.
Analytical solution of simple first order differential equations with separable variables.To include applications to practical problems.
Differentiation of simple functions defined implicitly or parametrically.The second derivative of curves defined implicitly or parametrically is not required.
Equations of tangents and normals for curves specified implicitly or in parametric form.
Simple cases of integration
using partial fractions.
Greatest level of difficulty

;

.

15.7 Vectors

Vectors in two and three dimensions.Column vectors will be used in questions but candidates may use notation if they wish.
Magnitude of a vector.
Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations. The result
Parallel vectors
Position vectors.
The distance between two points.
Vector equation of lines.Equations of lines in the form
Eg

To include the intersection of two straight lines in two and three
dimensions. Parallel lines. Skew lines in three dimensions.

The scalar product. Its use for calculating the angle between two lines.To include finding the coordinates of the foot of the perpendicular from a point to a line and hence the perpendicular distance from a point to a line.

16 Further Pure 1

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1 and Core 2.

Candidates will also be expected to know for section 16.6 that the roots of an equation can be located by considering changes of sign of in an interval of in which is continuous.

Candidates may use relevant formulae included in the formulae booklet without proof.

16.1 Algebra and Graphs

 

Graphs of rational functions of the form.

or

Sketching the graphs.

Finding the equations of the asymptotes which will always be parallel to the coordinate axes.

Finding points of intersection with the coordinate axes or other straight lines.

Solving associated inequalities.

Using quadratic theory (not calculus) to find the possible values of the function and the coordinates of the maximum or minimum points on the graph.

Eg for

which has real roots if , ie if ; stationary points are and

Graphs of parabolas, ellipses and hyperbolas with equations

and

Sketching the graphs.

Finding points of intersection with the coordinate axes or other straight lines. Candidates will be expected to interpret the geometrical implication of equal roots, distinct real roots or no real roots.

Knowledge of the effects on these equations of single transformations of these graphs involving translations, stretches parallel to the -axis or -axis, and reflections in the line .

Including the use of the equations of the asymptotes of the hyperbolas given in the formulae booklet.

16.2 Complex Numbers

Non-real roots of quadratic equations. Complex conjugates – awareness that non-real roots of quadratic equations with real coefficients occur in conjugate pairs.
Sum, difference and product of complex numbers in the form
Comparing real and imaginary parts. Including solving equations eg where is the
conjugate of .

16.3 Roots and coefficients of a quadratic equation

Manipulating expressions
involving and .
Eg
Forming an equation with roots or etc.

16.4 Series

Use of formulae for the sum of the squares and the sum of the cubes of the natural numbers. Eg to find a polynomial expression for

or

16.5 Calculus

Finding the gradient of the tangent to a curve at a point, by taking the limit as tends to zero of the gradient of a chord joining two points whose -coordinates differ by . The equation will be given as , where is a simple polynomial such as .
Evaluation of simple improper integrals. E.g.

16.6 Numerical Methods

Finding roots of equations by interval bisection, linear interpolation and the Newton-Raphson method. Graphical illustration of these methods.
Solving differential equations of the form Using a step-by-step method based on the linear approximations with given values for and .
Reducing a relation to a linear law. E.g.
Use of logarithms to base 10 where appropriate.
Given numerical values of , drawing a linear graph and using it to estimate the values of the unknown constants.

16.7 Trigonometry

General solutions of trigonometric equations including use of exact values for the sine, cosine and tangent of Eg ,

16.8 Matrices and Transformations

and matrices; addition and subtraction, multiplication by a scalar. Multiplying a matrix by a matrix or by a matrix.
The identity matrix for a matrix.
Transformations of points in the plane represented by matrices. Transformations will be restricted to rotations about the origin, reflections in a line through the origin, stretches parallel to the -axis and -axis, and enlargements with centre the origin.
Use of the standard transformation matrices given in the formulae booklet.
Combinations of these transformations
e.g.

17 Further Pure 2

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof except where proof is required in this module and requested in a question.

17.1 Roots of Polynomials

The relations between the roots and the coefficients of a polynomial equation; the occurrence of the non-real roots in conjugate pairs when the coefficients of the polynomial are real.

17.2 Complex Numbers

The Cartesian and polar coordinate forms of a complex number, its modulus, argument and conjugate. and .
The sum, difference, product and quotient of two complex numbers. The parts of this topic also included in module Further Pure 1 will be examined only in the context of the content of this module.
The representation of a complex number by a point on an Argand diagram; geometrical illustrations.
Simple loci in the complex plane.

For example,

Maximum level of difficulty where and are complex numbers.

17.3 De Moivre's Theorem

De Moivre's theorem for integral . Use of and , leading to, for example, expressing in terms of multiple angles and in term of powers of .
Applications in evaluating integrals, for example,.
De Moivre's theorem; the roots of unity, the exponential form of a complex number. The use, without justification, of the identity
Solutions of equations of the form To include geometric interpretation and use, for example, in expressing in surd form.

17.4 Proof by Induction

Applications to sequences and series, and other problems. Eg proving that is divisible by 6, or where n is a positive integer.

17.5 Finite Series

Summation of a finite series by any method such as induction, partial fractions or differencing. Eg

17.6 The calculus of inverse trigonometrical functions

 

Use of the derivatives of as given in the formulae booklet.

To include the use of the standard integrals

given in the formulae booklet.

17.7 Hyperbolic Functions

Hyperbolic and inverse hyperbolic functions and their derivatives; applications to integration. The proofs mentioned below require expressing hyperbolic functions in terms of exponential functions.

To include solution of equations of the form .

Use of basic definitions in proving simple identities.
Maximum level of difficulty:

.

The logarithmic forms of the inverse functions, given in the formulae booklet, may be required. Proofs of these results may also be required.
Proofs of the results of differentiation of the hyperbolic functions, given in the formula booklet, are included.
Knowledge, proof and use of:

Familiarity with the graphs of

.

17.8 Arc length and Area of surface of revolution about the x-axis

Calculation of the arc length of a curve and the area of a surface of revolution using Cartesian or parametric coordinates. Use of the following formulae will be expected:

18 Further Pure 3

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

18.1 Series and Limits

Maclaurin series
Expansions of ,
and , and for rational values of
Use of the range of values of for which these expansions are valid, as given in the formulae booklet, is expected to determine the range of values for which expansions of related functions are valid;

eg. .

Knowledge and use, for , of as tends to infinity and as tends to zero.
Improper integrals.

E.g..

Candidates will be expected to show the limiting processes used.

Use of series expansion to find limits. E.g.

18.2 Polar Coordinates

Relationship between polar and Cartesian coordinates.The convention    wil be used. The sketching of curves given by equations of the form     may be required. Knowledge of the formula   is not required.
Use of the formula
.

18.3 Differential Equations

The concept of a differential equation and its order. The relationship of order to the number of arbitrary constants in the general solution will be expected.
Boundary values and initial conditions, general solutions and particular solutions.

18.4 Differential Equations - First Order

Analytical solution of first order linear differential equations of the form where and are functions of .

To include use of an integrating factor and solution by complementary function and particular integral.

Numerical methods for the solution of differential eqations of the form .

Euler's formula and extensions to second order methods for this first order differential equation. Formulae to be used will be stated explicitly in questions, but candidates should be familiar with standard notation such as used in

Euler's formula

the formula ,

and the formula where and .

18.5 Differential Equations - Second Order

Solution of differential equations of the form

, where


, and are integers, by using an auxiliary equation whose roots may be real or complex.
Including repeated roots.
Solution of equations of the form


where , and are integers by finding the complementary function and a particular integral

Finding particular integrals will be restricted to cases where is of the form or a polynomial of degree at most 4, or a linear combination of any of the above.
Solutions of differential equations of the form:

where and are functions of . A substitution will always be given which reduces the differential equation to a form which can be directly solved using the other analytical methods in 18.4 and 18.5 of this specification or by separating variables.
Level or difficulty as indicated by:

(a) Given use the substitution

to show that .

Hence find in terms of
Hence find in terms of

(b)  use the subsitution

to show that

and hence that , where is an arbitrary constant.

Hence find in terms of .

19 Further Pure 4

Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

19.1 Vectors and Three-Dimensional Coordinate Geometry

Definition and properties of the vector product.

Calculation of vector products.

Including the use of vector products in the calculation of the area of a triangle or parallelogram.
Calculation of scalar triple products. Including the use of the scalar triple product in the calculation of the volume of a parallelepiped and in identifying coplanar vectors. Proof of the distributive law and knowledge of particular formulae is not required.
Applications of vectors to two- and three-dimensional geometry, involving points, lines and planes. Including the equation of a line in the form .
Vector equation of a plane in the form or .

Intersection of a line and a plane.

Angle between a line and a plane and between two planes.

Cartesian coordinate geometry of lines and planes. Direction ratios and direction cosines. To include finding the equation of the line of intersection of two non-parallel planes.

Including the use of where are the direction cosines.

Knowledge of formulae other than those in the formulae booklet will not be expected.

19.2 Matrix Algebra

Matrix algebra of up to 3 x 3 matrices, including the inverse of a 2 x 2 or 3 x 3 matrix.

Including non-square matrices and use of the results

and

Singular and non-singular matrices.

The identity matrix for 2 x 2 and 3 x 3 matrices.

Matrix transformations in two dimensions: shears.

Candidates will be expected to recognise the matrix for a shear parallel to the or axis. Where the line of invariant points is not the or axis candidates will be informed that the matrix represents a shear. The combination of a shear with a matrix transformation from MFP1 is included.
Rotations, reflections and enlargements in three dimensions, and combinations of these.

Rotations about the coordinate axes only.
Reflections in the planes only.

Invariant points and invariant lines.

Eigenvalues and eigenvectors of 2 x 2 and 3 x 3 matrices.

Characteristic equations. Real eigenvalues only. Repeated eigenvalues may be included.

Diagonalisation of 2 x 2 and 3 x 3 matrices.

where is diagonal matrix featuring the eigenvalues and is a matrix whose columns are the eigenvectors.
Use of the result

19.3 Solution of Linear Equations

Consideration of up to three linear equations in up to three unknowns.
Their geometrical interpretation and solution.

Any method of solution is acceptable.

19.4 Determinants

Second order and third order determinants, and their manipulation. Including the use of the result , but a general treatment of products is not required.
Factorisation of determinants. Using row and/or column operations or other suitable methods.
Calculation of area and volume scale factors for transformation representing enlargements in two and three dimensions.

19.5 Linear Independence

Linear independence and dependence vectors.