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Further Mathematics · 9231

Further Pure Mathematics 1

01Roots of Polynomial Equations

Roots and coefficients3

Roots, factors and multiplicity Deriving quadratic root relations Cubic and quartic root relations

Expressions in roots2

Evaluating expressions in roots Power sums from the original equation

New equations5

Shifting and scaling roots Reciprocal root transformations Equations for squared roots Equations for cubed roots Combined roots and chapter review

Reference1

Formula and method reference

02Rational Functions

Rational graphs4

Domains, intercepts and excluded points Horizontal asymptotes and first sketches Stationary points and quadratic denominators Oblique asymptotes and parameters

Range and inequalities2

The range of a rational function Solving rational inequalities

Graph transformations4

Reciprocal graphs Output modulus and input modulus Modulus graphs: asymptotes and solutions Graphs of y² = f(x)

Review1

Rational functions: review and method reference

03Summation of Series

Finite sums4

Sigma notation and linear sums Sums of squares and cubes Polynomial and patterned sums Changing summation limits

Differences and limits3

The method of differences Recognising more complex differences Convergence, remainders and bounds

Review1

Summation of series: method selection

04Matrices 1

Matrix algebra6

Matrix entries and operations Matrix multiplication Inverse matrices of order 2 Determinants Inverse matrices of order 3 Matrix equations and inverse products

Plane transformations5

Matrices and point images Stretches, enlargements and shears Rotations and reflections Combining matrix transformations Area scale factors

Invariant points and lines2

Invariant points Invariant lines

Review1

Matrices 1: mixed problems

05Polar coordinates

Coordinates and curves8

Polar coordinates and points Polar and Cartesian equations Tracing polar curves Symmetry and cardioids Petal curves Spirals Intersections of polar curves Greatest distances

Areas5

Areas from polar sectors Areas of petals Areas between curves Common regions Unfamiliar polar integrals

Review1

Polar coordinates review

06Vectors

Vector products3

Vectors and projection The vector product Areas using vectors

Lines and planes4

Lines and their intersections Point-to-line distance Equations of planes Plane forms and construction

Intersections and angles3

Line-plane intersections Intersection of two planes Angles with lines and planes

Distances and perpendiculars3

Point-to-plane distance Distance between skew lines Common perpendiculars

Review2

Vectors — Formula Summary Vectors review

07Proof by induction

Proof by induction8

Why induction works Proving sums Proving recurrence formulas Proving bounds Proving derivative patterns Proving matrix powers Proving divisibility Induction review

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Resources

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Further Pure Mathematics 1/Rational Functions

Solving rational inequalities

Use interval signs and endpoint conditions to compare rational expressions safely.

Contents

Further Mathematics · 9231

Further Pure Mathematics 1

01Roots of Polynomial Equations

Roots and coefficients3

Roots, factors and multiplicity Deriving quadratic root relations Cubic and quartic root relations

Expressions in roots2

Evaluating expressions in roots Power sums from the original equation

New equations5

Shifting and scaling roots Reciprocal root transformations Equations for squared roots Equations for cubed roots Combined roots and chapter review

Reference1

Formula and method reference

02Rational Functions

Rational graphs4

Domains, intercepts and excluded points Horizontal asymptotes and first sketches Stationary points and quadratic denominators Oblique asymptotes and parameters

Range and inequalities2

The range of a rational function Solving rational inequalities

Graph transformations4

Reciprocal graphs Output modulus and input modulus Modulus graphs: asymptotes and solutions Graphs of y² = f(x)

Review1

Rational functions: review and method reference

03Summation of Series

Finite sums4

Sigma notation and linear sums Sums of squares and cubes Polynomial and patterned sums Changing summation limits

Differences and limits3

The method of differences Recognising more complex differences Convergence, remainders and bounds

Review1

Summation of series: method selection

04Matrices 1

Matrix algebra6

Matrix entries and operations Matrix multiplication Inverse matrices of order 2 Determinants Inverse matrices of order 3 Matrix equations and inverse products

Plane transformations5

Matrices and point images Stretches, enlargements and shears Rotations and reflections Combining matrix transformations Area scale factors

Invariant points and lines2

Invariant points Invariant lines

Review1

Matrices 1: mixed problems

05Polar coordinates

Coordinates and curves8

Polar coordinates and points Polar and Cartesian equations Tracing polar curves Symmetry and cardioids Petal curves Spirals Intersections of polar curves Greatest distances

Areas5

Areas from polar sectors Areas of petals Areas between curves Common regions Unfamiliar polar integrals

Review1

Polar coordinates review

06Vectors

Vector products3

Vectors and projection The vector product Areas using vectors

Lines and planes4

Lines and their intersections Point-to-line distance Equations of planes Plane forms and construction

Intersections and angles3

Line-plane intersections Intersection of two planes Angles with lines and planes

Distances and perpendiculars3

Point-to-plane distance Distance between skew lines Common perpendiculars

Review2

Vectors — Formula Summary Vectors review

07Proof by induction

Proof by induction8

Why induction works Proving sums Proving recurrence formulas Proving bounds Proving derivative patterns Proving matrix powers Proving divisibility Induction review

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