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Subjects

CCambridge EPearson Edexcel AQAAQA OCROCR

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Mathematics · 9709

Pure Mathematics 3

01Algebra

Modulus5

Modulus as Distance and Equations Modulus Equations with Restrictions Equations with Two Moduli Graphs and Modulus Intersections Transforming Modulus Graphs and Ranges

Modulus Inequalities4

Modulus Inequalities Modulus Inequalities with a Variable Side Comparing Two Moduli Sums of Moduli

Polynomial Division3

Polynomial Division Division by Non-monic Linear Polynomials Division by Quadratic Polynomials

Factors and Remainders5

Factor and Remainder Theorems Finding Unknown Polynomial Coefficients Solving Polynomial Equations Quadratic Factors and Unknown Roots Polynomial Equations with Changed Inputs

02Logarithmic and exponential functions

Meaning and Graphs4

What a Logarithm Means Natural Logarithms and Calculator Keys Exponential and Logarithmic Graphs Shifted Functions and Their Inverses

Logarithm Laws and Equations6

The Logarithm Laws Combining and Expanding Logarithms Solving Logarithmic Equations Quadratic Logarithmic Equations Unknown Bases and Nested Logarithms Simultaneous Logarithmic Equations

Exponential Equations and Inequalities5

Solving Exponential Equations Equations with Repeated Powers Exponential Inequalities Exponential Models and Thresholds Exponential Graph Intersections

Linearising Relationships4

Linearising Power Models Linearising Exponential Models Choosing Transformed Axes Fitting Models to Logarithmic Data

03Trigonometry

Reciprocal Functions5

Reciprocal Trig Ratios Reciprocal Trig Graphs Transforming Reciprocal Trig Graphs Solving Reciprocal Trig Equations Pythagorean Trig Equations

Compound and Double Angles6

Compound-angle Formulae Exact Values with Compound Angles Equations with Compound Angles Double-angle Formulae Half-angle Values Equations with Double Angles

Identities and Equations3

Further Trig Identities Multiple-angle Identities Using Identities in Equations

R-form6

R-form Choosing Sine or Cosine R-form Solving R-form Equations Ranges and Extrema of R-form Squared and Reciprocal R-form Hidden R-form

04Differentiation

Choosing and Combining Rules3

Choosing a Differentiation Rule Using the Product Rule Using the Quotient Rule

Exponential, Logarithmic and Trigonometric Derivatives6

First Derivatives of eˣ and ln x Composite Exponential Functions Composite Logarithmic Functions Trigonometric Derivatives Inverse-tangent Derivative Mixed Rules and Stationary Points

Implicit Curves2

What Implicit Differentiation Means Implicit Curve Gradients

Parametric Curves3

What Parametric Differentiation Means Parametric Tangents and Normals Parametric Special Tangents and Direction

05Integration

Direct Antiderivatives3

Integration as reverse differentiation Integrating a reciprocal linear function Integrating sin, cos and sec²

Trigonometric Rewriting1

Integrating sin² and cos²

Definite Integrals and Area1

Definite Integrals and Area

Recognising Advanced Forms2

Inverse-tangent integrals and mixed numerators Recognising derivative over function

Transforming Integrals3

Integration using a given substitution Integrating after partial fractions Integration by parts

Choosing and Applying Methods1

Choosing an Integration Method

06Numerical Solution of Equations

Locating Roots1

Locating a root

Fixed-Point Iteration2

From an equation to an iteration Running and confirming an iteration

Applying Numerical Methods1

Using iteration in a longer problem

07Vectors

Vector Foundations4

Vector language and operations Displacement vectors from points Position vectors and constructing points Magnitude and unit vectors

Scalar Product and Geometry1

Scalar product and vector angles

Vector Lines4

Vector equation of a line Angle between two lines Parallel, intersecting and skew lines Foot of a perpendicular from a point

08Further Algebra

Partial Fractions2

Equal-degree fractions before partial fractions Choosing and solving partial fractions

Binomial Series3

Binomial series for rational powers Adapting a binomial series and its validity Partial fractions with binomial series

09Differential equations

Meaning and Solving3

What a differential equation means Solving a separable differential equation Complete solutions and integration choices

Forming Models2

Forming an equation from a rate statement Linked rates and net-rate models

Growth and Interpretation2

Growth, decay and limiting values Interpreting a differential-equation model

10Complex numbers

Complex Foundations2

Imaginary numbers and Cartesian form Complex arithmetic and the conjugate

Argand Geometry and Form2

Argand diagrams, modulus and argument Polar form, multiplication and division

Equations and Loci3

Complex roots and equations Standard loci in the complex plane Locus intersections and extreme values

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Resources

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Pure Mathematics 3/Integration

Integration by parts

Practise Integration

Contents

Mathematics · 9709

Pure Mathematics 3

01Algebra

Modulus5

Modulus as Distance and Equations Modulus Equations with Restrictions Equations with Two Moduli Graphs and Modulus Intersections Transforming Modulus Graphs and Ranges

Modulus Inequalities4

Modulus Inequalities Modulus Inequalities with a Variable Side Comparing Two Moduli Sums of Moduli

Polynomial Division3

Polynomial Division Division by Non-monic Linear Polynomials Division by Quadratic Polynomials

Factors and Remainders5

Factor and Remainder Theorems Finding Unknown Polynomial Coefficients Solving Polynomial Equations Quadratic Factors and Unknown Roots Polynomial Equations with Changed Inputs

02Logarithmic and exponential functions

Meaning and Graphs4

What a Logarithm Means Natural Logarithms and Calculator Keys Exponential and Logarithmic Graphs Shifted Functions and Their Inverses

Logarithm Laws and Equations6

The Logarithm Laws Combining and Expanding Logarithms Solving Logarithmic Equations Quadratic Logarithmic Equations Unknown Bases and Nested Logarithms Simultaneous Logarithmic Equations

Exponential Equations and Inequalities5

Solving Exponential Equations Equations with Repeated Powers Exponential Inequalities Exponential Models and Thresholds Exponential Graph Intersections

Linearising Relationships4

Linearising Power Models Linearising Exponential Models Choosing Transformed Axes Fitting Models to Logarithmic Data

03Trigonometry

Reciprocal Functions5

Reciprocal Trig Ratios Reciprocal Trig Graphs Transforming Reciprocal Trig Graphs Solving Reciprocal Trig Equations Pythagorean Trig Equations

Compound and Double Angles6

Compound-angle Formulae Exact Values with Compound Angles Equations with Compound Angles Double-angle Formulae Half-angle Values Equations with Double Angles

Identities and Equations3

Further Trig Identities Multiple-angle Identities Using Identities in Equations

R-form6

R-form Choosing Sine or Cosine R-form Solving R-form Equations Ranges and Extrema of R-form Squared and Reciprocal R-form Hidden R-form

04Differentiation

Choosing and Combining Rules3

Choosing a Differentiation Rule Using the Product Rule Using the Quotient Rule

Exponential, Logarithmic and Trigonometric Derivatives6

First Derivatives of eˣ and ln x Composite Exponential Functions Composite Logarithmic Functions Trigonometric Derivatives Inverse-tangent Derivative Mixed Rules and Stationary Points

Implicit Curves2

What Implicit Differentiation Means Implicit Curve Gradients

Parametric Curves3

What Parametric Differentiation Means Parametric Tangents and Normals Parametric Special Tangents and Direction

05Integration

Direct Antiderivatives3

Integration as reverse differentiation Integrating a reciprocal linear function Integrating sin, cos and sec²

Trigonometric Rewriting1

Integrating sin² and cos²

Definite Integrals and Area1

Definite Integrals and Area

Recognising Advanced Forms2

Inverse-tangent integrals and mixed numerators Recognising derivative over function

Transforming Integrals3

Integration using a given substitution Integrating after partial fractions Integration by parts

Choosing and Applying Methods1

Choosing an Integration Method

06Numerical Solution of Equations

Locating Roots1

Locating a root

Fixed-Point Iteration2

From an equation to an iteration Running and confirming an iteration

Applying Numerical Methods1

Using iteration in a longer problem

07Vectors

Vector Foundations4

Vector language and operations Displacement vectors from points Position vectors and constructing points Magnitude and unit vectors

Scalar Product and Geometry1

Scalar product and vector angles

Vector Lines4

Vector equation of a line Angle between two lines Parallel, intersecting and skew lines Foot of a perpendicular from a point

08Further Algebra

Partial Fractions2

Equal-degree fractions before partial fractions Choosing and solving partial fractions

Binomial Series3

Binomial series for rational powers Adapting a binomial series and its validity Partial fractions with binomial series

09Differential equations

Meaning and Solving3

What a differential equation means Solving a separable differential equation Complete solutions and integration choices

Forming Models2

Forming an equation from a rate statement Linked rates and net-rate models

Growth and Interpretation2

Growth, decay and limiting values Interpreting a differential-equation model

10Complex numbers

Complex Foundations2

Imaginary numbers and Cartesian form Complex arithmetic and the conjugate

Argand Geometry and Form2

Argand diagrams, modulus and argument Polar form, multiplication and division

Equations and Loci3

Complex roots and equations Standard loci in the complex plane Locus intersections and extreme values

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