New Syllabus Mathematics for O-Level 1

164 Pages·1979·15.842 MB·English

by  Owen Perry B.Sc., Joyce Perry B.Sc. (auth.)

New Syllabus Mathematics for 0-Level 1
New Syllabus Mathematics for 0-Level 1
New Syllabus Mathematics for 0-Level 2
New Syllabus Mathematics for 0-Levell
Owen Perry, B.Sc.,
Head of Department of General and
Professional Studies,
Lewes Technical College
Joyce Perry, B.Sc.,
Formerly Mathematics Teacher,
Lewes High School
M
© Owen Perry and Joyce Perry 1979
All rights reserved. No part of this publication may be
reproduced or transmitted, in any form or by any means,
without permission.
First published 1979 by
THE MACMILLAN PRESS LTD
London and Basingstoke
Associated companies in Delhi Dublin
Hong Kong Johannesburg Lagos Melbourne
New York Singapore and Tokyo
British Library Cataloguing in Publication Data
Perry, Owen William
New syllabus mathematics for 0-Level.
1.
1. Mathematics~1961~
I. Title II. Perry, Joyce
510 QA39.2
ISBN 978-1-349-03505-2 ISBN 978-1-349-03503-8 (( eeBBooookk))
DOI 10.1007/978-1-349-03503-8
The paperback edition of this book is sold subject to the
condition that it shall not, by way of trade or otherwise, be
lent, resold, hired out, or otherwise circulated without the
publisher's prior consent in any form of binding or cover
other than that in which it is published and without a
similar condition including this condition being imposed
on the subsequent purchaser.
Contents
Preface vii
Notation viii
1. Number Systems 1
Natural numbers, multiples and factors, prime numbers, counting in
bases other than ten. Operations defined on the natural numbers,
associative, commutative and distributive laws. Integers and directed
numbers, rational numbers, decimal fractions, common fractions.
Irrational, real and complex numbers, conditional statements.
2. Arithmetic in Base Ten 17
Harder fractions, percentages, profit and loss, changing in a given
ratio, change of units, scales and maps, sharing in a given ratio, simple
interest. Standard index form, estimations and approximations, limits
of accuracy. Use of four figure mathematical tables.
3. Statistics ; Introduction to Probability 33
Populations and samples, tabulation, graphical representation of
statistics, bar, pie and sectional charts, line graphs. Frequency
distributions, discrete and continuous variables, histograms, freq
uency polygons, cumulative and relative cumulative frequency. Meas
ures of location, mean, mode, median. Measures of dispersion, range,
percentiles, mean deviation, standard deviation, Historigrams, moving
averages and trends. Probability, random selection, outcomes and
events.
4. Simple Algebra 53
Algebraic expressions, statements and equations. Multiplication and
division, addition and subtraction, factors, brackets and the distri
butive law. Fractions, solution of simple and simultaneous equations,
changing the subject of a formula, solution of algebraic problems.
5. Matrices 64
Size of a matrix, addition and subtraction, multiplication by a number,
matrix multiplication. Special matrices, zero, transpose, square,
symmetric, unit, inverse, singular. Determinant and inverse of 2-
square matrices. Uses of matrices, transformations of the plane,
solution of simultaneous equations, networks, incident and route
matrices. Problems involving matrices.
6. Introduction to Geometry; Area and Volume 85
Points, lines, planes, angles. Polygons, triangles, Pythagoras' theorem.
Quadrilaterals. Symmetrical properties of plane figures, line and point
symmetry of triangles and quadrilaterals. Mensuration of triangles
and quadrilaterals. Circles, circumference, area, length of arc, area of
sector, area of segment. Circular measure. Mensuration of geometrical
solids. Similarity, ratio of sides, areas and volumes of similar figures.
7. Sets and Further Probability 110
Set language and notation, element of a set, universal set, complement
of a set, subsets. Combination of sets, operations union and in
tersection. Venn diagrams, properties of union and intersection,
number of elements in a set, addition and product formulae. Further
probability; independent and dependent events, conditional pro
bability, use of Venn diagrams and tree diagrams.
8. Mappings and Functions 125
Domain arid range, function notation, arrow diagrams, intervals on
the real number line. Composition of functions, inverse functions.
Linear equations and inequalities. Graphs oflinear functions, gradient
and intercept. Graphical solution of linear equations and inequalities,
linear programming.
Answers to Exercises 142
Index 153
Preface
These two volumes are intended for students who want to pass 0-Level
mathematics in the modern syllabus. They are particularly suitable for those
who need to follow a thorough revision course, whether at school or as full
time, day-release or evening students at colleges offurther education. Since the
only mathematical knowledge assumed is simple arithmetic, the books are also
suitable for those who need a pass in 0-Level mathematics to improve their
promotion prospects, and are starting the modern syllabus for the first time.
The majority of the exercises are divided into A and B sections. The
questions in the A sections are generally shorter and intended for routine
practice in the techniques appropriate to each part of the text. Longer and
more thought-provoking questions are found in the B sections. Each of the
sixteen chapters ends with a multiple-choice test and a selection of mis
cellaneous examples from past examination papers.
The authors are grateful to Dr Patricia Dauncey, for her helpful criticism of
the manuscript and for working through the exercises. They also wish to thank
the Controller of H.M.S.O. for permissions to use Statistical Abstracts.
The text covers the 'modern' alternative syllabus of each of the major
examining boards, and the authors acknowledge with thanks the permission
given by the boards to quote examination questions. The source of each
question is shown in the text by the following abbreviations
(AEB) Associated Examining Board
(C) University of Cambridge Local Examinations Syndicate
(L) University of London, University Entrance and School
Examinations Council
(JMB) Joint Matriculation Board
(NI) Northern Ireland Schools Examinations Council
(0) Oxford Delegacy of Local Examinations
(OCSMP) Oxford and Cambridge Schools Examination Board.
School~ Mathematics Project
(S) Southern Universities Joint Board
(SCE) Scottish Certificate of Education Examination Board.
Notation
{a,b,c, .... } the set of a, b, c, ....
such that
E is an element of
!/' is not an element of
n( ) the number of elements in the. set of
0 the empty (null) set
the universal set
~
u union
n intersection
c is a subset of
A' the complement of the set A
N the set of natural numbers
z
the set of integers
R the set of real numbers
PQ operation Q followed by operation P
__f:x-_+y the function of mapping the set X into the set Y
f(x) the image of x under the function f
F' the inverse of the function f
fg the function f of the function g
-o---o-- open interval on the number line
closed interval on the number line
{x:-2<x<7} the set of values of x such that ...
= implies that
<= is implied by
implies and is implied by
is equal to
is identically equal to
is approximately equal to
is not equal to
is less than
is less than or equal to
is greater than
is greater than or equal to
is not less than
is not greater than
the unsigned part of a signed number, that is the modulus
CJJ infinity
M' the transpose of the matrix M
1
Number Systems
The two essentials of education are literacy and Prime Numbers
numeracy. To be literate requires a knowledge of the
letters of the alphabet, ordering these into words Every number has 1 as a factor and a natural
and constructing sentences according to the rules of number which has no factors except itself and 1 is
grammar; to be numerate requires a knowledge of called a prime number. The primes up to 100 are 2, 3,
numbers, order relations between the numbers, and 5, 7, 11, 13,17,19,23,29,31,37,41,43,47,53,59,61,
operations defined on the numbers according to 67, 71, 73, 79, 83, 89, 97.
certain laws. In this chapter the four different kinds Some mathematicians enjoy finding prime num
of numbers are defined, and the three laws which bers, and it has been proved that they form an
govern the basic operations of arithmetic are in infinite set. The largest one known up to 1971 is said
troduced; addition and subtraction, multiplication to have 6002 digits.
and division.
Example 1.1
Express as a product of prime factors (a) 35, (b) 90,
1.1 Natural Numbers (c) 144.
(a) 35 = 5 x 7
The counting numbers are called natural numbers. (b) 90 = 2 X 3 X 3 X 5
In the United Kingdom, as in most countries, the (c) 144 = 2 X 2 X 2 X 2 X 3 X 3
denary system of counting in tens is used, and
anyone starting a mathematics course for 0-level Highest Common Factor
must be familiar with the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0
as well as the addition and multiplication tables The highest common factor (H.C.F.) of a given set of
associated with them. No other prior knowledge will numbers is the greatest number which is a factor of
be assumed except the ordinary processes of each of the given numbers. Thus the H.C.F. of 15, 27,
arithmetic on the natural numbers including long 39 is 3, since
multiplication and division. 15 = 3 X 5, 27 = 3 X 3 X 3, and 39 = 3 X 13
3 is the greatest number which is a factor of each.
Example 1.2
Multiples and Factors
Find the H.C.F. of 42, 56, 126.
An even number is a multiple of 2 and the first even Expressing each number as a product of primes
number is 2 x 1, the next is 2 x 2, the third 2 x 3 and 42 = 2 X 3 X 7
so on. Similarly, 14 is 7 x 2, 21 is 7 x 3, and so 14 and 56= 2X 2 X 2 X 7
21 are both multiples of 7. 126 = 2 X 3 X 3 X 7
The converse of the statement '14 is a multiple of The primes which are factors of all three numbers
7' is '7 is a factor of 14', or '14 has 7 as a factor'. are 2 and 7 therefore the H.C.F. is 2 x 7 = 14.
Number Systems