Sixth Form Pure Mathematics. Volume 1

452 Pages·1968·16.007 MB·English

by  C. Plumpton and W. A. Tomkys (Auth.)

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Sixth Form
Pure Mathematics
VOLUME ONE
C. PIUMPTON
Queen Mary College, London
W. A. TOMKYS
Belle Vue Boys Grammar School
Bradford
SECOND EDITION
PERGAMON PRESS
OXFORD NEW YORK TORONTO SYDNEY
PARIS KRANKKURT
U. K. Pergamon Press Ltd., Headington Hill Hall,
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AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., 19a Boundary
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FEDERAL RE PUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus,
OF GERMANY Pferdstrasse 1, Federal Republic of Germany
Copyright © 1968 Pergamon Press Ltd.
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted in
any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publishers
First edition 1962
Reprinted with corrections 1965
Second edition 1968
Reprinted 1972, 1973, 1975, 1978
Library of Congress Catalog Card No. 67-30688
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
ISBN 0 08 009374 4 Flexi-cover
PREFACE TO THE SECOND EDITION
THIS book is the first of a series of volumes on Pure Mathematics and
Theoretical Mechanics for Sixth Form students whose aim is entrance
into British and Commonwealth Universities or Technical Colleges.
A knowledge of Pure Mathematics up to G.C.E. O-level is assumed
and the subject is developed by a concentric treatment in which each
new topic is used to illustrate ideas already treated. The major topics
of Algebra, Calculus, Coordinate Geometry and Trigonometry are
developed together.
This volume covers most of the Pure Mathematics required for the
single subject Mathematics at Advanced Level. Volume Two covers
the remainder, with the exception of Pure Geometry, of the Pure Mathe­
matics required for a double subject at Advanced Level. Early and rapid
progress in calculus is made at the beginning of this volume in order
to facilitate the student's progress along the most satisfactory lines in
Pure Mathematics, in Theoretical Mechanics and in Physics.
The worked examples are an essential feature of this book and they
are followed by routine exercises within the text of each chapter, asso­
ciated closely with the work on which they are dependent. The exercises
at the end of each chapter collectively embody all the topics of that
chapter and, where possible, the preceding chapters also. They are grad­
ed in difficulty and in all cases the last few of these exercises might
well be deferred, by most students, until a second reading. Most of these
miscellaneous exercises are taken from the examination papers of the
University of London (L.), the Northern Universities Joint Matricula­
tion Board (N.), the Oxford and Cambridge Schools Examination
Board (O.C.) and the Cambridge Local Examinations Syndicate (C).
We are grateful to the authorities concerned for permission to reprint
and use these questions.
ix
X PREFACE TO THE SECOND EDITION
The sections, exercises and equations are numbered according to
the chapters; e.g. § 5.4 is the fourth section of Chapter V; Ex. 9.5 is the
set of exercises at the end of § 9.5; equation (3.7) is the seventh (number­
ed) equation of Chapter m. Only those equations to which subsequent
references are made are numbered.
In this second edition we have made some minor changes and cor­
rections. We are grateful to all those who have made suggestions for
improvements and corrections. In particular we thank Mr. J. A. Croft
and Mr. G. Hawkes who read the proofs.
C. PLUMPTON
W. A. TOMKYS
CHAPTER I
INTRODUCTION TO THE CALCULUS
1.1 Coordinates and Loci
Coordinates. The position of a point in a plane may be defined by
reference to a pair of rectangular axes Ox, Oy with which the student
will be familiar from drawing the graphs of functions of x. Thus in
Fig. 1 (#1, yi) denotes the point whose x-value or x-coordinate is x\
and whose j-coordinate is y±. Negative jc-coordinates are by convention
to the left of the origin and negative ^-coordinates below the origin.
Fig. 1 shows the points A(3, 2), B(-l 2), C(-2, -1) and Z)(l, -2).
9
4
•(-1,2) *(3.2)
* *,
y.
y
0
C»(-2,-l)
D.(..-21
Fig. 1.
vi
2 SIXTH FORM PURE MATHEMATICS
The Equation of a Curve. The relation which is satisfied by the
coordinates of any point on a particular curve is called the equation of
the curve. Thus if the equation of the curve is y = f(x) [y is a function
of x] and if (xi, y{) is a point on the curve, then y± =f(x{). The curve
is the locus of a point which moves so that its coordinates always satisfy
the equation.
Example. If f(x) = ax2+bx+c and if the points (4, —9), ( — 1, —4),
9
(0, —1) lie on the curve y =/(*), calculate the values of a, b and c.
Since (4, —9), ( — 1, —4), and (0, —1) each lie on y = /(*), then
-9=/(4), -4=/(-l) and -l=/(0);
.-. -9=16a + 46 + c, -4 = a-6-fc, -l = c
from which
a= - 1, b = 2, c == - 1.
The Gradient of a Straight Line. The gradient of the straight line
joining the points P(a, b) and Q(p q) is defined as (q—b)/(p—a). It re­
9
presents the rate of change of y compared with x along PQ.
Exercises 1.1
1. Show on a diagram the positions of the following points whose coordinates
are given: (2,4), (5, -1), (-3, 2), (0,4), (5,0).
2. In each of the following, the equation of a curve is given together with one
coordinate of a point on the curve. Calculate, in each case, the possible values of
the other coordinate:
(i) y — JC!+5X+4, (* = 3)
(ii) x*+y* = 25, (y = 3)
(iii) y = logi„ x, 0- = 2)
(iv) y = x3-x, (y = 0)
(v) y = 5x+4, (x = a^- h).
3. (i) If /■<*) = 3^1 > calculate/(-l);
(ii) If f(x) = (xt+5x+l3)1", calculate/(2);
INTRODUCTION TO THE CALCULUS 3
(iii) If /(*) = — +4, calculate /( - 2);
(iv) If /(*) = log x, write down/(—j ;
10
(v) If /(*) = sin (2x+10)°, write down /(40).
4. Write down the gradients of the lines joining each of the following pairs of
points:
(i) (2,4), (-1,5);
(ii) (-2,3), (4, -2);
(iii) (-3, -2), (-5, -3);
(iv) (a, -*), (-«,«;
(v) (2/, 2//), (-2/f, -20.
b c
5. If f(x) = a+—+— and if f(x) = 3-5 when x = -2, /(*) = 7 when
t
JC = — 1, and f(x) = 16 when x = 0*5, calculate the values of a, b, and c.
/ 1 1\
6. Show that the gradient of the line joining (h, k) to I - —, —r-l is equal to
the gradient of the line joining (/i, k) to the origin and hence show that the three
points are collinear.
State which of the following sets of points are collinear:
(i) (3,5), (-2,1), (4,0);
(ii) (1,3), (-5,1), (7,5);
(iii) (2m, -/), (0,0), (/, m);
(iv) (2//, -1/m), (l//,0), (0, 1/m).
7. Calculate the gradient of the line joining the points whose x-coordinates are
1 and -2 respectively on the curve y — x+4/x. Show that this chord is parallel to
the chord joining the two points on the curve whose ^-coordinates are 4 and — \
respectively.
8. The curve y = f(x) cuts the jc-axis three times at x = 2, x = 3, and x = — 2.
If/(x) is a polynomial function of x of degree three of the form x3 + ax*+bx+c,
find/(x).
4 SIXTH FORM PURE MATHEMATICS
1.2 The Idea of a Limit
Consider the sum of the series of numbers 1+Y + J+-|-+ We
have:
i . X _ A
i -r 2 — 2 »
1 . JL . JL - .7.
1 T" 2 T- 4 — 4 »
1 4--1-4.i4.JL - 15
1 + 2 T 4 T8 — 8 >
The addition of each number halves the gap between the sum that has
been arrived at and 2. By taking sufficient numbers the sum can be
made as near to 2 as we please. In these circumstances 2 is defined as the
limit of the sum of the series as the number of terms increases indefinitely
(or as the number of terms tends to infinity) and this statement is written,
n 1 + + = 2
l L( 4 T -4) -
n
A similar result to this is illustrated by a recurring decimal. Thus by
division -|- = 0-1 and this is equivalent to the statement
. . / 1 11 1 \ 1
h?14To+Io-2+ios+ •• +io*) = T-
n
The tangent to a curve can be considered as the limiting position of a
chord as the two points of intersection approach one another. Thus in
Fig. 2 the tangent to the curve at P is the limiting position of the chord
PQ as Q approaches P.
x2—a2
Algebraic Limits. Consider lim . Direct substitution of
x2—a2
x = a in the expression gives an indeterminate result, but
x—a
x2 — a2
lim = lim (x-\-a)= 2a.
x-*-a ^ ^ x-*- a
INTRODUCTION TO THE CALCULUS
This result can be extended:
xn-an
lim if n is a positive integer
^ x-a
x a
= lim (xn-1+xn-2a+xn-*a2+. . , + a71-1) by division;
-. xn-an ,
n
.'. lim = na71"1. (1.1)
-+ x-a
x a
Fig. 2.
It can be proved, but it will not be proved here, that result (1.1) is true
also when n is fractional or negative. The result is illustrated by two
particular cases thus:
= l i m ^ - ^= (V^^)(V^+Va)
lim lim
^ x-a -+ (x-a)(y/x+y/a)
^ x-a x a x a
x a
i -\
lim
_1 1_
. x~x—a" x a .. 1 1
flim = lim = lim =—=.
_* x-a ^ ax a2
x a x a