# Sixth Form Pure Mathematics

### Volume 2

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Sixth Form Pure Mathematics, Volume 2, provides an introduction to inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, and a range of mathematical methods including the use of determinants, the manipulation of inequalities, the solution of easy differential equations, and the use of approximate numerical methods. Complex numbers are defined and the various ways of representing and manipulating them are considered. Polar coordinates, curvature, an elementary study of lengths of curves and areas of surfaces of revolution, a more mature discussion of two-dimensional coordinate geometry than was possible in Volume 1, and an elementary introduction to the methods of three dimensional coordinate geometry comprise the geometrical content of the book. Throughout, the authors have tried to preserve the concentric style which they used in Volume 1 and the many worked examples and exercises in each chapter are designed or chosen to provide a continuous reminder of the work of the preceding chapters. Except for Pure Geometry, the two volumes cover almost all of the syllabuses for Advanced Pure Mathematics of the nine Examining Boards. This book provides an adequate course for mathematical pupils at Grammar Schools and a useful introductory course for Science and Engineering students in their first year at University or Technical College or engaged in private study.

## Table of Contents

Chapter XI Linear Equations and Determinants

The Solution of Linear Simultaneous Equations in Two Unknowns

Second Order Determinants

Rules of Manipulation for Second Order Determinants

Third Order Determinants

Factorization of Determinants

Geometrical Interpretation.

The Equation of a Straight Line through Two Given Points

The Equation of a Tangent to the Curve

The Equation of a Line-Pair

The Equation of a Circle through Three Given Points

The Area of a Triangle

The Solution of Simultaneous Equations

Summary

The Product of Two Determinants

The Derivative of a Determinant

Chapter XII Inverse Circular Functions, Hyperbolic Functions and Inverse Hyperbolic Functions

Inverse Circular Functions.

The Derivatives of the Inverse Circular Functions

Standard Integrals

The Hyperbolic Functions

Inverse Hyperbolic Functions

Derivatives of the Inverse Hyperbolic Functions

Logarithmic Forms of the Inverse Hyperbolic Functions

Methods of Integration

The Integrals

Summary of Standard Integrals and Methods of Integration So Far Considered

Chapter XIII Definite Integrals. Further Applications of Integration

Properties of Definite Integrals

Infinite Integrals

Reduction Formula

Approximate Numerical Integration

Mean Values and Root Mean Square

Center of Mass

The Theorem of Pappus concerning Volumes

Moments of Inertia

Chapter XIV Some Properties of Curves

Points of Inflexion

The Length of a Curve

The Cycloid

Areas of Surfaces of Revolution

The Theorem of Pappus concerning Surfaces of Revolution

Curvature

Newton's Formula for Radius of Curvature at the Origin

Chapter XV Polar Coordinates

Definitions

Loci in Polar Coordinates

Curve Sketching in Polar Coordinates

The Lengths of Chords of Polar Curves Which are Drawn through The Pole

Transformations from Polar to Cartesian Equations and the Reverse Process

Areas in Polar Coordinates

The Length of an Arc in Polar Coordinates

Volumes of Revolution and Areas of Surfaces of Revolution in Polar Coordinates

The Angle between the Tangent and the Radius Vector

The Tangential Polar Equation—Curvature

Chapter XVI Complex Numbers

The Number System

Definition of Complex Number

The Cube Roots of Unity

Conjugate Pairs of Complex Roots

The Geometry of Complex Numbers

The Polar Coordinate Form of a Complex Number- Modulus and Argument

Products and Quotients

De Moivre's Theorem

The Exponential Form of a Complex Number

Exponential Values of Sine and Cosine

Chapter XVII Differential Equations

Formation of Differential Equations

The Solution of a Differential Equation

First Order Differential Equations with Variables Separable

Homogeneous Equations

The Law of Natural Growth

Linear Equations of the First Order

Bernoulli's Equation

Equations of Higher Orders

Linear Equations of the Second Order with Constant Coefficients

The Complementary Function. The Particular Integral

Chapter XVIII Approximate Numerical Solution of Equations

Graphical Methods

The Number of Real Roots of An Equation

The Approximate Value of a Small Root of a Polynomial Equation

Newton's Method for Obtaining a Closer Approximation to a Real Root of An Equation

Chapter XIX Inequalities

Rules of Manipulation

Fundamental Inequalities

The Calculus Applied to Inequalities

Chapter XX Coordinate Geometry

The Straight Line

Lone Pairs

The Circle

The Radical Axis

Coaxal Circles

Conic Sections

Note on the General Equation of the Second Degree

The Chord of Contact of Tangents Drawn from an External Point to a Conic-Pole and Polar

The Equation of a Pair of Tangents Drawn from an External Point to a Conic

Conjugate Diameters

The Polar Equation of a Conic

Chapter XXI Coordinate Geometry of Three Dimensions

The Coordinate System

The Distance between Two Points

The Coordinates of a Point Which Divides the Line Joining Two Given Points in a Given Ratio

The Equation of a Plane

The Equations of a Line—Direction Cosines

The Angle between Two Directions

The Intersection of Three Planes

Answers to Exercises

Index

## Product details

- No. of pages: 416
- Language: English
- Copyright: © Pergamon 1963
- Published: January 1, 1963
- Imprint: Pergamon
- eBook ISBN: 9781483140889