Advanced Level Mathematics: Pure Mathematics 2 and 3

336 Pages·2002·5.9 MB·English

by  Hugh Neill

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BCAMBRIDGE
• UNIVERSITY PRESS
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The publishers would like to acknowledge the contributions of the following people
to this series of books: Tim Cross, Richard Davies, Maurice Godfrey, Chris Hockley,
Lawrence Jarrett~ David A. Lee, Jean Matthews, Norman Morris, Charles Parker,
,;:9
Geoff Staley, Rex Stephens, Peter Thomas and Owen Toller.
,,~
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo,
New Delhi
Cambridge University Press
The Edinburgh Building, Cambridge CB2 2RU, UK
www.cambridge.org
Information on this title: www.cambridge.org/9780521530125
© Cambridge University Press 2002
I,•
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
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First published 2002
4th printing 2005
First South Asian edition 2006
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Reprinted 2007, 2008, 2009 (twice)
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Printed in India by Replika Press Pvt. Ltd.
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A catalogue record for this publication is available from the British Library
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r ISBN-13: 978 0 521 69635 7 paperback
This edition is for sale in South and South East Asia only, not for export elsewhere.
Cover image: © D. Boone/CORBIS
Contents
Introduction iv
Unit P3 UnitP2
-
~ Polynomials 3
~'2 The modulus function 18
-.-3 Exponential and logarithmic functions 30
. 4 Differentiating exponentials and logarithms 50
v5 Trigonometry 65
,6 Differentiating trigonometric functions 82
Revision exercise 1 97
._l_.-Diffe rentiating products 99
~ Solving equations numerically 108
~e trapezium rule 122
-10- Parametric equations 130
'U----Clilves defined implicitly 141
Revision exercise 2 153
Practice examinations for P2 155
~ectors: dimensi~ns
lines in two and three 161
ectors: planes in three dimensions 175
c
$The binomial expansion 188
,15"' Rational functions 198
~Complex numbers 223
\J7 Complex numbers in polar form 241
~Integration 258
19 Differential equations 275
Revision exercise 3 296
Practice examinations for P3 300
Answers 305
Index 329
/
Introduction
Cambridge International Examinations (CIE) Advanced Level Mathematics has been
created especially for the new CIE mathematics syllabus. There is one book
corresponding to each syllabus unit, except for this book which covers two units, the
second and third Pure Mathematics units, P2 and P3.
The syllabus content is arranged by chapters which are ordered so as to provide a viable
teaching course. The first eleven chapters are required for unit P2; all the chapters are
required for unit P3. This is indicated by the vertical grey bars on the contents page.
A few sections include important results that are difficult to prove or outside the syllabus.
These sections are marked with an asterisk(*) in the section heading, and there is usually a
sentence early on explaining precisely what it is that the student needs to know.
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I Some paragraphs within the text appear in this type style. These paragraphs are usually
~\I outside the main stream of the mathematical argument, but may help to give insight, or
1'
r suggest extra work or different approaches.
\.
Graphic calculators are not pe_rmitted in the examination, but they are useful aids in learning
/'
t . mathematics. In the book the authors have noted where access to a graphic calculator would
l be especially helpful but have not assumed that they are available to all students.
I
Numerical work is presented in a form intended to discourage premature approximation.
In ongoing calculations inexact numbers appear in decimal form like 3.456 ... , signifying ·
that the n'umber is held in a calculator to more places than are given. Numbers are not
rounded at this stage; the full display could be, for example, 3.456123 or 3.456 789.
Final answers are then stated with some indication that they are approximate, for
t . example ' 1.23 correct to 3 significant figures_'.
There are plenty of exercises, and each chapter ends with a Miscellaneous exercise
which includes some questions of examination standard. There are two Revision
exercises for the material common to units P2 and P3, and a further Revision exercise
f for unit P3. There 'are also two Practice examination papersfor unitP2 at the end of
P2&3, and two Practice examination papers for unit P3 at the end of P3.
I Some exercises foclude questions that go beyond the likely requirements of the
I examinations, either in difficulty or in length or both. In the P2&3 chapters some
I questions may be more appropriate for P3 than for P2 students. Questions marked with
I
I an asterisk require knowledge of results or techniques outside the syllabus.
Cambridge University Press would like to thank OCR (Oxford, Cambridge and RSA
Examinations), part of the University of Cambridge Local Examinations Syndicate (UCLES)
group~ for permission to use past examination questions set in the United Kingdom.
The authors tha\}k UCLES and Cambridge University Press, in particular Diana Gillooly,
for their help in producing this book. However, the responsibility for the text, and for any
errors, remains·with the authors.
Unit P2 and Unit P3
The subject content of unit P2 is a subset of the subject content of unit P3.
This part of the book (pages 1-158) comprises the subject content ofunit P2,
and is required for both units P2 and P3. The additional material required to
complete unit P3 is contained in the second part of the book. ·
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1 Polynomials
This chapter is about polynomials.which include linear and quadratic expressions.
When you have completed it, you should
• be able to add, subtract, multiply and divide polynomials
• understand the words 'quotient' and 'remainder' used in dividing polynomials
• be able to use the method of equating coefficients
• be able to use the remainder theorem and the factor theorem.
1.1 Polynomials
You already know a good deal about polynomials from your work on quadratics in
Chapter 4 of Pure Mathematics 1 (unit Pl), because a quadratic is a special case of a
polynomial. Here are some examples of polynomials.
3x3-2x2+1 3 4-2x x2
2x4 1-2x+3x5 -J2x2 lxl7 x
2
A (non-zero) polynomial, p(x), is an expression in x of the form
axn +bxn-I + ... + jx+k
where a, b, c, ... ,k are real numbers, a :;e 0, and n is a non-negative integer.
The number n is called the degree of the polynomial. The expressions axn, bxn-I , ... , jx
and k which make up the polynomial are called terms. The numbers a, b, c, ... ,j and k are
called coefficients; a is the leading coefficient. The coefficient k is the constant term.
Thus, in the quadratic polynomial 4x2 - 3x + 1 , the degree is 2; the coefficients of x2
and x, and the constant term, are 4, - 3 and 1 respectively.
Polynomials with low degree have.special names: ifthe polynomial has
• degree 0 it is called a constant polynomial, or a constant
• degree 1 it is called a linear polynomial
• degree 2 it is called a quadratic polynomial, or a quadratic
• degree 3 it is called a cubic polynomial, or a cubic
• degree 4 it is called a quartic polynomial, or a quartic.
When a polynomial is written ~s a.xn + bxn-I + ... + jx + k, with the term of highest
degree first and the other terms in descending degree order finishing with the constant
term, the terms are said to be in descending order. If the terms are written in the reverse
order, they are said to be in ascending order (or ascending powers of x). For example,
3x4 + x2 - 1x + 5 is in descending order; in ascending order it is 5-1x + x2 + 3x4. It is
the same polynomial whatever order the terms are written in.
4 PURE MATHEMATICS 2 & 3
The functions _!_ = x-1 and -JX = x+ are not polynomials, because the powers of x are
x
not positive integers or zero.
Polynomials have much in common with integers. You can add them, subtract them and
multiply them together and the result is another polynomial. You can even divide a
polynomial by another polynomial, as you will see in Section 1.4.
1.2 Addition, subtraction and multiplication of polynomials
To add or subtract two polynomials, you simply add or subtract the co(,!fficients of
corresponding powers; in other words, you collect like terms. Suppose that you want to
add 2x3 + 3x2 - 4 to x2 - x - 2. Then you can set out the working like this:
2x3 + 3x2 - 4
x2 - x - 2
2x' + 4x2 - x - 6
Notice that you must leave gaps in places where the coefficient is zero. You need to do
addition so often that it is worth getting used to setting out the work in a line, thus:
(2x3 + 3x2 -4) + (x2 - x -2) = (2 + O)x3 + (3 + l)x2 + (0 + (-l))x + ((-4) + (-2))
= 2x3 + 4x2 - x - 6.
You will soon find that you can miss out the middle step and go straight to the answer.
The result of the polynomial calculation (2x3 + 3x1 - 4 )-(2 x3 + 3x2 - 4) is 0. This is
a special case, and it is called the zero polynomial. It has no degree.
Look back at the definition of a polynomial, and see why the zero polynomial was not
included there.
Multiplying polynomials is harder. It relies on the rules for multiplying out brackets,
a(b+c+ ... + k) =ab +ac+ ... +ak and (b +c+ ... +k)a = ba+ca+ ... +ka.
To apply these rules to multiplying the two polynomials 5x + 3 and 2x2 - 5x + 1 ,
replace 2x2 - 5x + 1 for the time being by z. Then
(5x + 3)(2x2 - 5x + 1) = (5x + 3)z
= 5xz+3z
= sx( 2x2 - 5x + 1) + 3( 2x2 - 5.x' + 1)
= (10x3 -25x2 + sx) +(6x2 -15x + 3)
= 10x3 -19x2 - lOx + 3.
In practice, it is easier to note that every term in the left bracket multiplies every term in
the right bracket. You can show this by setting out the steps in the following way.
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CHAPTER 1: POLYNOMIALS 5
2x2 - 5x + 1 x
10x3 - 25x2 + 5x 5x
+ 6x2 - 15x + 3 +3
10x3 + (-25+6)x2 + (5-15)x + 3
giving the result 10x3 -19x2 - lOx + 3.
It is worth learning to work horizontally. The arrows below show the term 5x from the
first bracket multiplied by -5x from the second bracket to get -25x2.
(5x+3)(2x2-5~5x+ 1)+~2x2-5x+ 1)
= (10.X -25x2 + 5x)+ (6x2 -15x +3)
=10x3-"-19x2 -10x+3.
You could shorten the process and write
(5x + 3)( 2x2 - 5x + 1) = 10x3 - 25x2 + 5x + 6x2 -15x + 3
= 10x3 -19x2 - lOx + 3.
If you multiply a polynomial of degree m by a polynomial of degree n, you have a
calculation of the type
( ax m + bx m-1 +... )(Axn + B x n-1 + ... ) =aA xm+n + ...
in which the largest power of the product is m + n . Also the coefficient aA is not zero
because neither of a and A is zero. This shows that:
Exercise lA
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1 State the degree of each of the following polynomials.
(a) x3-3x2+2x-7 (b) 5x+l (c) 8+5x-3x2 +7x+6x4
(d) 3 (e) 3-5x (f) XO
2 In each part firid p(x) + q(x), and give your answer in descending order.
(a) p(x)=3x2+4x-1, q(x)=x2+3x+7
(b) p(x)=4x3+5x2-7x+3, q(x)=x3-2x2+x-6
(c) p(x)=3x4-2x3+7x2-1, q(x)=-3x-x3+5x4+2
(d) p(x) = 2-3x3 +2x5, q(x) = 2x4 +3x3-5x2+1
(e) p(x) = 3+ 2x-4x2 -x3, q(x) = 1-7x+2x2
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6 PlJRE MATHEMATICS 2 & 3
3 For each of the pairs of polynomials given in Question 2 find p(x)-q(x).
4 Note that p(x) + p(x) may be shortened to 2p(x). Let p(x) = x3 - 2x2 + 5x - 3 and
q(x) = x2 -x + 4. Express each of the following as a single polynomial.
(a) 2p(x)+q(x) (b) 3p(x)-q(x) (c) · p(x)-2q(x) (d) 3p(x)-2q(x)
5 Find the following polynomial products.
(a) (2x--:: 3)(3x +l) (b) (x2 +3x-l)(x-2)
(c) (x2 +x-3)(2x+3) (d) (3x-1)(4x2 -3x+2)
(e) (x2 +2x-3)(x2 +1) (f) (2x2 -3x+1)(4x2 +3x-5)
(g) (x3 +2x2 -x+6)(x+3) (h) (x3 -3x2 +2x-l)(x2 -2x-5)
(i) (1+3x-x2 +2x3)(3-x+2x2) (j) ( 2 - 3x + x2 )( 4 - 5x + x3)
(k) (2x + 1)(3x-2)(x + 5) (1) (x2 +l)(x-3)(2x2-x+l)
6 In each of the following products find the coefficient of x and the coefficient of x2•
(a) (x+2)(x2 -3x+6) (b) (x-3)(x2 +2x-5)
(c) (2x+ 1)(x2 -5x+l) (d) (3x-2)(x2 '--2x+7) \
(e) (7y -3)(3x2 -6x+l) (f) (2x-5)(3x3 -x2 +4x+2)
(I!\ (x2-L2x-3)(x2+3x-4) (h) (3x2+1)(2x2-5x+3)
(i) (x2 +3x-l)(x3+x2-2x+l) (j) (3x2-x+2)(4x3-5x+l)
7 In each of the following the product of Ax+ B with another polynomial is given. Using the
fact that A and B are constants, find A and B.
(a) (Ax+B)(x-3)=4x2 -llx-3 (b) (Ax+B)(x+5)=2x2 +7x-.15
(c) (Ax+ B)(3x-2) = 6x2 -x-2 (d) (Ax+ B)(2x + 5) = 6x2 + llx-10
(e) (Ax+ B)(x2 -1) = x3 +2x2 -x-2 (f) (Ax+B)(x2 +4)=2x3-3x2 +8x-12
(g) (Ax+ B)(2x2 -3x+4) = 4x3-x+12 (h) (Ax+B)(3x2-2x-1)=6x3-7x2+1
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1.3 Equations and idegtities
In this chapter so far you have learned how to add, subtract and multiply polynomials,
and you can now carry out calculations such as
(2x + 3) + (x - 2) = 3x + 1,
( x2 - 3x - 4 )-(2x + 1) = x2 - 5x - 5 and
(1-x)(l+ x+ x2) = l-x3
fairly automatically.
However, you .should realise that these are not equations in the normal sense, because
they are true for all values of x.
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