Advanced mathematics 1

384 Pages·1980·57.603 MB·English

by  C W Celia, A T F Nice, K F Elliott (auth.), Dr C Plumpton (eds.)

Advanced
mathematics
1
C W Celia
formerly Principal Lecturer in Mathematics,
City of London Polytechnic
AT F Nice
Mathematics Department, Lady Eleanor Holies
School, Hampton; formerly Principal Lecturer in
Mathematics, Middlesex Polytechnic
K F Elliott
formerly Head of the Division of Mathematics Education,
Derby Lonsdale College of Higher Education
Consultant Editor: Dr C Plumpton,Jormerly
Reader in Engineering Mathematics, Queen Mary College,
London and Moderator in Mathematics,
University of London School Examinations Board
REVISED EDITION
Macmillan Education
© C. W. Celia, A. T. F. Nice & K. F. Elliott 1980
All rights reserved. No reproduction, copy or transmission
of this publication may be made without written permission.
No paragraph of this publication may be reproduced, copied
or transmitted save with written permission or in accordance
with the provisions of the Copyright Act 1956 (as amended).
Any person who does any unauthorised act in relation to
this publication may be liable to criminal prosecution and
civil claims for damages.
First published 1980
Reprinted 1982, 1983 (twice)
Revised edition 1985
Reprinted 1986
Published by
MACMILLAN EDUCATION LTD
Houndmills, Basingstoke, Hampshire RG21 2XS
and London
Companies and representatives
throughout the world
British Library Cataloguing in Publication Data
Celia, Cuthbert Walter
Advanced mathematics. I
I. Mathematics-1961-
1. Title II. Nice, A. T. Ill. Elliott, K. F.
510 QA39.2
ISBN 978-0-333-23192-0
ISBN 978-0-333-39983-5 ISBN 978-1-349-08303-9 (eBook)
DOl 10.1007/978-1-349-08303-9
Contents
Preface v
1 Sets. functions and operations
Sets Intersection and union of sets Ordered pairs Functions
Composition of functions Inverse functions Operations
2 Algebra 1 19
The graph of y = ax2 +bx +c Relations between the roots and the
coefficients of equations Linear inequalities Quadratic inequal-
Ities Inequalities involving the modulus sign Location of the
roots of an equation
3 Algebra 2 49
Sequences Series Series of powers of the natural numbers
Mathematical induction Geometric progressions Infinite
geometric progressions The binomial expansion The remainder
and factor theorems Partial fractions The use of partial fractions
with expansions The use of partial fractions in the summation of
series
4 Trigonometry 1 82
Angles of any magnitude The graphs of y = sin x, y = cos x,
y = tan x Solution of equations General solution of equations
Relations between trigonometrical ratios Circular measure The
graphs of y = a sin (bx +c) and y = a cos (bx +c) Further solution
of equations
5 Trigonometry 2 105
The sine formula The cosine formula Addition theorems
a cos x + b sin x Factor formulae Area of a triangle Problems
in two or three dimensions Length of an arc and area of a sector of a
circle The circular functions
6 Differentiation 1 140
The idea of a limit Gradients Derived functions Rates of
change Differentiation of sin x and cos x The product rule The
quotient rule The chain rule Differentiation of inverse functions
7 Coordinate geometry 163
Lengths and angles Division of a line segment in a given ratio
Area of a triangle The straight line Distance of a point from a
line The circle x2 + y2 = r2 The circle (x-a)2 + (y-b)2 = r2
The parabola y2 = 4ax Parameters Polar coordinates Curves
of the form r = f(O)
8 Real and complex numbers 206
Number systems The natural numbers (positive integers) The
integers The rational numbers The real numbers The complex
numbers Geometrical representation of complex numbers (the
Argand diagram) Modulus and argument of a complex number
Trigonometric and polar forms of a complex number The triangle
inequality for complex numbers
9 Differentiation 2 230
Maximum and minimum values Points of inflexion Implicit
differentiation CurVe sketching Small increments
10 Integration 250
The indefinite integral Integration by substitution Integration by
parts The definite integral Change of limits The area under a
curve Mean values The definite integral as the limit of a sum
Volumes of solids of revolution
11 Natural logarithms and the exponential function 275
Indices and logarithms Natural logarithms Integrals involving
logarithms Logarithmic differentiation The exponential
function Maclaurin series The exponential series The
logarithmic series Separable differential equations
12 Vectors 309
Free vectors Vector addition Position vectors The ratio
theorem The vector equation of a straight line Unit vectors
Scalar products Differentiation of vectots Vectors in three
dimensions
13 Matrices 333
Order of a matrix Addition and subtraction of matrices Matrix
multiplication Square matrices The inverse of a matrix Linear
transformations of the x-y plane
Notation 351
Formulae 354
Answers 359
Index 377
iv Contents
Preface
This is the first of a series of books written for students preparing for A level
Mathematics. Books 1 and 2 of the series cover the work required for a single
subject A level in Mathematics or in Pure and Applied Mathematics. Book I
covers the essential core of sixth-form mathematics now accepted by the GCE
Boards, while Book 2 covers the applied mathematics, i.e. the numerical
methods, mechanics and probability, contained in most single-subject syl
labuses. Book 3 covers the additional pure mathematics needed by students
taking the double-subject Mathematics and Further Mathematics, and by those
taking Pure Mathematics as a single subject.
Vector notation and vector techniques are used wherever appropriate,
particularly where these methods illuminate or simplify the work, but their use is
avoided whenever they appear likely to confuse the student. Set language is
employed wherever it is considered helpful, but it is not introduced at all times as
a matter of principle.
The material is arranged under well-known headings and is organised so that
the teacher is free to follow his or her own preferred order of treatment. The
chapter contents are listed and an index is also provided to make it easy for both
the teacher and the student to refer back rapidly to any particular topic. For ease
of reference, a list of the notation used is given at the back of the book together
with a list of formulae.
Each topic is developed mainly through worked examples. There is a brief
introduction to each new piece of work followed by worked examples and
numerous simple exercises to build up the student's technical skills and to
reinforce his understanding. It is hoped that this will enable the individual
student working on his own to make effective use of the books, and the teacher to
use them with mixed ability groups. At the end of each chapter there are many
miscellaneous examples, taken largely from past A level examination papers. In
addition to their value as examination preparation, these miscellaneous
examples are intended to give the student the opportunity to apply the
techniques acquired from the exercises throughout the chapter to a considerable
range of problems of the appropriate standard.
We are most grateful to the University of London University Entrance and
School Examinations Council (L), the Associated Examining Board (AEB), the
University of Cambridge Local Examinations Syndicate (C) and the Joint
Matriculation Board (JMB) for giving us permission to use questions from their
past examination papers.
We are also grateful to the staff of Macmillan, particularly Mrs Janet Hawkins
and Mr Tony Feldman, for the patience they have shown and the help they have
given us in the preparation of these books.
C. W. Celia
A. T. F. Nice
K. F. Elliott
vi Preface
1 Sets, functions and
operations
1.1 Sets
All manner of people collect things ranging from match-boxes and beer mats to
works of art of great value. Many collectors sort their collections into categories,
e.g. stamps sorted by country of origin.
Mathematicians have this habit of collecting and sorting. From earliest time
men have been interested in numbers and have arranged them in sets, e.g. even
numbers, prime numbers, etc. Indeed the idea of a set is so important in
mathematics that a notation and a language have been developed for dealing with
sets.
A set is a well-defined collection of objects called elements or members of the
set.
Well-defined means that objects which are members ofa set are distinguishable
from objects which are not, and each member of a set is different from every other
member of the set. The fair-haired people in a room would not be a well-defined
collection since in some cases it might be a matter of opinion whether a particular
person was fair or not. Also the numbers 1, 1, 2 would not constitute a set.
The elements of a set may be named in a list or may be given by a description,
and the list or description is enclosed in braces { }. For instance, the set of days of
the week may be given as
{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
or as {the days of the week}.
Sets may be finite, i.e. having a finite number of elements, or they may be
infinite. The days of the week form a finite set and the set of even numbers is an
infinite set.
In addition to the braces used above, other symbols have been devised to enable
statements about sets to be made with brevity and precision. For instance, let
E = {even numbers}. The statement '2 is an element of the set of even numbers'
or '2 belongs to the set of even numbers' or indeed '2 is an even number' may be
abbreviated to
2 e E, e standing for 'is an element or or 'belongs to'.
Similarly the statement '3 is not an element of the set of even numbers' or '3 is not
an even number' may be abbreviated to
3 fl E, ¢ standing for 'is not an element or or 'does not belong to'.
C. W. Celia et al. Advanced mathematics 1 © C.W. Celia, A.T.F. Nice & K.F. Elliott 1980
The colon: is used as an abbreviation for 'such that', e.g. {x:x2- 9 = 0}
which is read as 'the set of all values of the number x such that x2 - 9 = 0'.
If each member of set A belongs to set Band also each member of set B belongs
to set A, then A and B are the same set.
Exercises 1.1
1 Define (i) by listing the elements, (ii) by describing the elements, four
different sets of
(a) numbers (b) shapes (c) non-mathematical objects.
2 (a) Give two examples of a finite set of numbers.
(b) Give two examples of an infinite set of numbers.
3 List the elements of the sets
(a) {prime numbers less than thirty}
(b) {letters of the alphabet consisting of straight lines only}
(c) {perfect squares between 2 and 50}.
4 Describe in words the sets
(a) {2, 4, 6, 8} (d) {x:x > 0}
(b) {2, 3, 5, 7, 11} (e) {x:x ~ 0}
(c) {x:x2 =1} (f) {x:O<x<3}.
5 If A = { 1, 2, 3, 4, 5} and B = { 8, 9, 10 }, complete the following statements by
inserting E, fi, A, B or elements of the sets A and B:
(a)3 ... A (c)2E ... (e)1¢ ... (g)5 ... B (i) ... ¢A
(b) 4 ... B (d) 9E... (f) 8~... (h) 8 ... A (j) ... ~B.
6 Given that A = {factors of 30}, express in set notation the statements
(a) 2 is a factor of 30 (b) 4 is not a factor of 30.
Express in words the statements (c) 5 EA (d) 7 ~A.
1.2 Intersection and union of sets
It is possible for an element to belong to two or more sets.
Let A = { 1, 2, 3, 4, 5} and B = { 2, 4, 6 }. Then 2 E A and 2 E B. This is illustrated
in Fig. 1.1 in a diagram which is known as a Venn diagram. The numbers 2 and 4
~
A~B
Fig. 1.1
belong to both set A and set B. { 2, 4} is the set of numbers belonging to both set A
and set B. This set is represented by An B and is called the intersection of sets A
and B. An B is read as 'A intersection B' or sometimes as 'A cap B'. The set An B
consists of those elements which belong to both set A and set B,
2 Advanced mathematics r
i.e. 2eA and 2eB and so 2e(AnB),
1 EA but 1 ¢B and so 1 ¢(An B).
Let P = {X : X > 1 } and Q = {X : X < 4}.
Then P n Q = { x : 1 < x < 4}.
Two sets which have no common elements are said to be disjoint. Consider the
disjoint sets E = {even numbers} and D = {odd numbers}. There are no
numbers which belong to both sets and so there are no elements in the set En D.
The set with no elements is called the empty set and is represented by 0 or { }.
Thus En D = 0 or En D = { }.
Consider again the sets A = { 1, 2, 3, 4, 5} and B = { 2, 4, 6 }. The numbers 1, 2, 3,
4, 5, 6 all belong to either the set A or the set B or to both set A and set B. This set
{ 1, 2, 3, 4, 5, 6} is known as the union of the sets A and B and is represented by
Au B. Au B is read as 'A union B' or sometimes as 'A cup B'.
In Fig. 1.2 the shaded region in Venn diagram (a) represents the set Au Band
the shaded region in Venn diagram (b) represents the set An B.
(a) (b)
Fig. 1.2
Figure 1.3 shows the Venn diagram for the disjoint sets E and D.
Fig. 1.3
Sometimes all the elements of one set are members of another set. For instance,
let N = {0, 1, 2, 3, ... }. Then every element of the set E = {positive even
numbers} belongs to the set N. The set E is then said to be a subset of the set N
and this is written as E c N and is illustrated in the Venn diagram in Fig. 1.4.
"I'
E ~
( )
N [Jj]
._)"
Fig. 1.4
Sets. functions and operations 3