Further Mathematics

1. The Triangle

1.1. Medians

1.2. Circumcentre and Circumcircle

1.3. Orthocentre and Pedal Triangle

1.4. Nine-point Circle

1.5. Euler Line

1.6. Incentre and Excentres

1.7. Concurrence and Collinearity

1.8. The Theorems of Ceva and Menelaus

1.9. Simson's Line

2. Algebraic Methods

2.1. Basic Identities

2.2. Important Factors

2.3. Remainder Theorem

2.4. Factors of Symmetrical Expressions

2.5. Highest Common Factor of Two Polynomials

2.6. Ratio and Proportion

2.7. Equal Fractions

2.8. Quadratic and Rational Quadratic Functions

2.9. Partial Fractions — Special Methods

2.10. Miscellaneous Equations

3. Determinants. Linear Equations

3.1. Notation and Definitions

3.2. Laws of Determinants

3.3. Factorisation of Determinants

3.4. Minors and Cofactors

3.5. Linear Simultaneous Equations

3.6. Equations in Two and Three Unknowns

3.7. Special Cases

3.8. Homogeneous Linear Equations in Two and Three Unknowns

4. More Advanced Analytical Geometry of the Straight Line and Circle

4.1. Applications of Determinants

4.2. The Line Pair

4.3. Line Pair Through the Origin

4.4. Angles Between Two Straight Lines

4.5. Bisectors of the Angles Between a Line Pair

4.6. Tangents to a Circle from an External Point

4.7. Chord of Contact of Tangents from an External Point to a Circle

5. Systems of Circles

5.1. Power of a Point with Respect to a Circle

5.2. Radical Axis of Two Circles

5.3. Radical Centre of Three Circles

5.4. Simplified Form for the Equations of Two Circles

5.5. Coaxal Systems of Circles

5.6. The Equation S₁ + λS₂ = 0

5.7. Simplified Form of the Equation of a Coaxal System

5.8. Types of Coaxal Systems

5.9. Limiting Points

5.10. Orthogonal Circles

5.11. Condition that Two Circles Should Cut Orthogonally

5.12. Orthogonal Systems

5.13. The Circle of Apollonius

5.14. Apollonius Circles as a System of Coaxal Circles

6. Complex Numbers

6.1. Definition, Notation and Geometrical Representation of a Complex Number

6.2. Argand Diagram

6.3. Modulus and Amplitude

6.4. Fundamental Processes

6.5. Equality, Addition and Subtraction

6.6. Inequalities

6.7. Multiplication and Division

6.8. Geometrical Properties of the Argand Diagram

6.9. Transformations

6.10. Exponential Form of a Complex Number

7. Elementary Theory of Equations

7.1. Polynomials and Polynomial Equations

7.2. Basic Theorems

7.3. Complex and Irrational Roots

7.4. Zeros of a Polynomial

7.5. Rolle's Theorem

7.6. Conditions for Two Equations to Have a Common Root

7.7. Repeated Roots

7.8. Solutions of Equations with Repeated Roots

7.9. Relations Between the Roots and Coefficients of a Cubic Equation

7.10. Quartic Equation and Equation of the nth Degree

7.11. Sums of Powers of Roots of an Equation

7.12. Transformation of Equations

8. The Conics. Miscellaneous Problems

8.1. Chord of Contact of Tangents from a Point to a Conic

8.2. Polar and Pole

8.3. Parallel Chords of a Conic

8.4. Equation of a Chord with a Given Mid-point

8.5. Normals to a Conic

8.6. Concyclic Points on a Conic

8.7. General Equation of a Conic

8.8. Systems of Conics, S + λS′ = 0

8.9. System of Conics Passing Through Four Given Points

8.10. Special Cases of the System S + λS′ = 0

9. De Moivre's Theorem and Simple Applications

9.1. De Moivre's Theorem

9.2. Values of (cos θ + i sin θ)^(1/q)

9.3. Binomial Equations

9.4. Factors of xⁿ − 1

9.5. Factors of xᵐ + 1

9.6. Factors of x²ⁿ − 2xⁿ cos nθ + 1

9.7. Expressions for cosⁿ θ and sinⁿ θ in Terms of Multiple Angles

9.8. Expansions of cos nθ, sin nθ and tan nθ

10. Finite Series. Mathematical Induction

10.1. Finite Series

10.2. Series Related to the Arithmetic Series

10.3. Trigonometrical Series

10.4. Applications of De Moivre's Theorem

10.5. Power Series

10.6. Binomial Series and Coefficients

10.7. Mathematical Induction

11. Expansions

11.1. Infinite Series

11.2. Expansion of a Function

11.3. Algebraic and Trigonometrical Methods

11.4. Taylor's Theorem

11.5. Maclaurin's Theorem

11.6. Expansion by Differentiation or Integration of a Known Series

11.7. Expansion by the Formation of a Differential Equation

11.8. Leibnitz's Theorem

12. Systematic Integration. Properties of Definite Integrals

12.1. Basic Theorems of Integration

12.2. Important Algebraic Integrals

12.3. Integrals Involving the Function √(ax² + bx + c)

12.4. Trigonometrical Integrals

12.5. Hyperbolic Integrals

12.6. Reduction Formulae

12.7. General Theorems on Definite Integrals

12.8. The Logarithmic Function ∫₁ˣ dt/t

12.9. Infinite Integrals: Infinite Range

12.10. Infinite Integrals: Infinite Integrand

13. Further Properties of Plane Curves

13.1. Curves Expressed in Polar Coordinates

13.2. Pedal Equation

13.3. Arc Length and Area of Surface of Revolution

13.4. Polar Form of Radius of Curvature

13.5. Radius of Curvature — Pedal Form

13.6. Area of a Closed Curve

13.7. Theorems of Pappus

13.8. Volume of Revolution of a Sectoral Area

14. Algebraic Geometry of the Plane and Straight Line

14.1. Rectangular Cartesian Coordinates in Three Dimensions

14.2. Three-dimensional Loci

14.3. Fundamental Results

14.4. Direction of a Straight Line

14.5. Direction-cosines

14.6. Angle Between Two Straight Lines

14.7. Direction Ratios

14.8. The General Equation of the First Degree

14.9. Perpendicular Form of the Equation of a Plane

14.10. Perpendicular from a Point to a Plane

14.11. Important Results on the Plane

14.12. The Straight Line

14.13. Symmetrical Form of the Equations of a Straight Line

14.14. Coplanar Lines

14.15. Skew Lines

14.16. Common Perpendicular to Two Skew Lines

14.17. Simplified Form for the Equations of Two Skew Lines

14.18. Some Properties of the Tetrahedron

15. The Sphere

15.1. The Equation of a Sphere

15.2. Sphere Passing Through Four Points

15.3. Tangent Plane to a Sphere

15.4. Length of Tangent Lines to a Sphere from a Given Point

15.5. Plane Section of a Sphere

15.6. Common Points of Two Spheres

15.7. Equations of a Circle

15.8. Equation of a Sphere Passing Through a Given Circle

15.9. System of Spheres Passing Through the Common Points of Two Spheres

16. Partial Differentiation

16.1. Functions of More than One Variable

16.2. Definition of a Partial Derivative

16.3. Geometrical Interpretation of Partial Derivatives in the Case of a Function of Two Variables

16.4. Higher Partial Derivatives

16.5. Small Changes

16.6. Differentials

16.7. Total Differential Coefficient

16.8. Important Special Cases

16.9. Derivatives of Implicit Functions

16.10. Total Partial Derivative

16.11. Homogeneous Functions

16.12. Euler's Theorem

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