SMP AS/A Mathematics Core 4 for AQA The School Mathematics Project
SMP AS/A Mathematics writing team Spencer Instone, John Ling, Paul Scruton, Susan Shilton, Heather West SMP design and administration Melanie Bull, Pam Keetch, Nicky Lake, Cathy Syred, Ann White The authors thank Sue Glover for the technical advice she gave when this AS/A project began and for her detailed editorial contribution to this book The authors are also very grateful to those teachers who advised on the book at the planning stage and commented in detail on draft chapters CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB RU, UK wwwcambridgeorg Information on this title: wwwcambridgeorg/9780560504 The School Mathematics Project 005 First published 005 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN- ISBN-0 978-0-5-6050-4 paperback 0-5-6050-X paperback Typesetting and technical illustrations by The School Mathematics Project The authors and publisher are grateful to the Assessment and Qualifications Alliance for permission to reproduce questions from past eamination papers Individual questions are marked AQA NOTICE TO TEACHERS It is illegal to reproduce any part of this work in material form (including photocopying and electronic storage) ecept under the following circumstances: (i) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence eists, or where you wish to eceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter of the Copyright, Designs and Patents Act 988, which covers, for eample, the reproduction of short passages within certain types of educational anthology and reproduction for the purposes of setting eamination questions
Contents Rational epressions 6 A Simplifying 6 factorising numerator and denominator, cancelling factors, division by an algebraic fraction B Adding and subtracting 9 use of lowest common denominator C Etension: Leibniz s harmonic triangle application of techniques from sections A and B to some fraction patterns D Etension: the harmonic mean 4 application of techniques from sections A and B Rational epressions 7 A Using the remainder theorem 7 polynomial divided by epression of form (a b), factor theorem to identify factors of the numerator B Further division 9 converting improper fraction to linear or quadratic epression and proper fraction C Further addition and subtraction further cases where the lowest common denominator is not the product of the denominators D Partial fractions 4 using simultaneous equations; finding numerator constant by substituting into the identity a value of chosen to eliminate the other numerator constants E Further partial fractions 8 repeated linear term in denominator Parametric equations A Coordinates in terms of a third variable plotting parametrically defined curve, simple geometric transformations, points of intersection with the aes B Converting between parametric and cartesian equations 5 eliminating parameter by first making it the subject of one parametric equation, or by equating a simple function of the parameter C Circle and ellipse 9 a cos T, y b sin T,with circle (a b r) as special case, obtaining cartesian equation from parametric equations 4 The binomial theorem 46 A Reviewing the binomial theorem for positive integers 46 B Etending the binomial theorem 47 ( a) n for n a negative integer or a fraction, applying the condition ÔaÔ< C Multiplying to obtain epansions 5 epansion of rational epression D Adding (using partial fractions) to obtain epansions 5 Mied questions 54 5 Trigonometric formulae 56 A Addition formulae 56 use of formulae for sin (A 8 B), cos (A 8 B) and tan (A 8 B), double angle formulae and their application to half-angles B Equivalent epressions 6 changing an epression of the form a sin b cos into one of the form r sin ( A) or r cos ( A), solution of equations in a given interval
6 Differential equations 68 Key points from Core 68 A Integration revisited 68 use of other variables than B Forming a differential equation 69 first order differential equation from a practical problem, growth and decay C Solving by separating variables 70 general solution, boundary conditions, particular solution D Eponential growth and decay 74 P Ae bt, limiting value, use of eponential functions as models E Further eponential functions 76 conversion of function involving a into one involving e Mied questions 78 7 Differentiation 80 Key points from previous books 80 A Functions defined parametrically 80 dy dy use of dt ; gradient, tangent d d dt and normal for parametrically defined curve B Functions defined implicitly 84 differentiation, with respect to, ofepressions involving both and y; gradient, tangent and normal for implicitly defined curve Mied questions 88 8 Integration 90 A Using partial fractions 90 indefinite integration of proper and improper algebraic fractions B Definite integrals 94 using partial fractions C Using trigonometric identities 95 using double angle and addition formulae Mied questions 97 9 Vectors 98 A Vectors in two dimensions 98 magnitude (modulus) and direction, equal vectors, addition and subtraction of vectors, multiplication of vector by scalar, geometrical interpretation of operations on vectors, parallel vectors B Components in two dimensions 0 column vectors, linear combination of vectors, unit vector, i-, j- notation C Vectors in three dimensions 05 etension of ideas in sections A and B to three dimensions, Pythagoras s theorem to find magnitude of vector in three dimensions D Position vectors in two and three dimensions vector between two points defined by position vectors, distance between two points in three dimensions E The vector equation of a line 5 the form a tb where t is scalar parameter and b is direction vector F Intersecting lines point of intersection in two and three dimensions from vector equations by solving simultaneous equations, skew lines, determining whether lines intersect G Angles and the scalar product 5 angle between two vectors from cost ab, a b scalar product 0 for perpendicular lines, H The angle between two straight lines 8 concept of the angle between skew lines, using scalar product to find angle between skew or intersecting lines I Shortest distance finding foot of perpendicular from point to a line and perpendicular distance Mied questions 6 Answers 8 Inde 8
Rational epressions In this chapter you will learn how to simplify rational epressions add, subtract, multiply and divide rational epressions A Simplifying (answers p 8) An epression that consists of one polynomial divided by another is called a rational epression or algebraic fraction n A A function is defined by f(n) where n is an integer such that n ^ 0 n 4n (a) Evaluate each of these in its simplest fractional form (b) (i) (c) (i) (i) f() (ii) f(4) (iii) f(0) Without calculating, what do you think is the value of f(00) in its simplest form? (ii) Check your result What do you think is the value of f(k) in its simplest fractional form? (ii) Prove your conjecture (d) Show that f(n) % for all positive integer values of n A A function is defined by f(n) n 6n 5 where n is an integer such that n ^ 0 n 7n 0 (a) Evaluate each of these in its simplest fractional form (b) (i) (i) f(0) (ii) f() (iii) f(0) What do you think is the value of f(k) in its simplest fractional form? (ii) Prove your conjecture (c) Hence show that the equation f(n) \ has no integer solution 5 0 A Prove that, when is a multiple of 5, the value of the epression can be written as a unit fraction (with as its numerator) When simplifying rational epressions, it is usually beneficial to factorise whenever possible Eample Simplify - - 4 Solution Factorise Divide numerator and denominator by ( 4) - - 4-4 -4 6 Rational epressions
Eample Simplify n n-5 n n 5 Solution Factorise Divide numerator and denominator by (n 5) n - n - n n-5 n n 5 n n-5 n n 5 ( ) ( n 5) ( n- ) ( n ) ( n 5) You may need to revise dividing by a fraction 5 For eample, ) % % 9% % 5 5 K In general, dividing by a fraction is equivalent to multiplying by its reciprocal This rule applies to all rational epressions: a ) c a d b d b c Eample 4n 4 Simplify ) 8 n - 9 n 5n- Solution Use the rule for dividing 4n 4 8 4 4 ) n n 5n n - 9 n 5n- n - 9-8 4( n ) n n Factorise ( n ) ( n- ) ( - ) 8 ( n ) ( n- ) Divide numerator and denominator by 4 and (n ) n - Eercise A (answers p 8) Simplify each of these (a) 6-6 - (b) (c) (d) (e) 5-5 (a) Show that 4-4 is equivalent to - 7 (b) Simplify each of these 5n -0 (i) (ii) - n 7 n - n 7 - n 6n 4n 4n n Rational epressions 7
Simplify each of these (a) 5 6 5 (b) - - 4 (c) (d) (e) 6 --8-5 4 A function is defined by g(n) n 8 where n is a positive integer n 0n 9 (a) Evaluate g() and g(4) in their simplest fractional form (b) When n is odd, prove that in its simplest form g(n) is a unit fraction (c) Find the value of n such that g(n) ( 5 Simplify each of these (a) n 9n 0 n 5n-4 n -0n- (b) (c) n n 0 n - 7n 0 n 6n 5 (d) (e) n n- n 7n-4 n n (f) (g) n 7n 6 n - 48 n - n (h) 6 Simplify each of these (a) (c) (e) 7 Simplify each of these (a) (c) -0-5 7 5-0 6 9 - ) 4 5 5-5 ) - 0-4 8 Show that ) y is equivalent to the single fraction y 9 Simplify each of these (b) (d) (f) (b) (d) (a) ) (b) ) (c) 6 ) 4 (d) ) (e) 5 ) 0 y y 0 Functions are defined by f() 6 ( $ 0) and g() ( $ ) - (a) Evaluate fg(7) (b) Show that fg() ( ) -6-4 6-0 5 5-4 4 4-9 - 5 5-5 - ) - 4 6 4 9-4 ) 6-7 -4-70 9-5 n - 6n 5 n -n-0 n n - n 9n - 5n-4 n - 9n 8 Rational epressions
Functions are defined by f() 6 8 and g() 0 8 where is a positive integer (a) (i) Evaluate f() and g() in their simplest fractional form f (ii) Find in its simplest form g (b) Prove that f g is always an integer B Adding and subtracting Fractions can easily be added or subtracted if they are written with the same denominator The lowest common multiple of two denominators is called the lowest common denominator and is usually the simplest to use 9 8 For eample, - : D Algebraic fractions can be dealt with in the same way 4 4 Eample 4 Epress as a single fraction in its simplest form - - 5 Solution A suitable denominator is 5( ) 5 - - 5 5 ( - ) - ( - ) 5-5-( -) 5 ( - ) 5 6 Epand the brackets in the numerator - 5 ( - ) Simplify 6 5 ( - ) ( ) Factorise if possible 5- Eample 5 Epress 4 Solution as a single fraction in its simplest form Write 4 as a fraction with a denominator of Epand the brackets and simplify 4 4 4 ( ) 8 7 Rational epressions 9
Eample 6 Epress as a single fraction in its simplest form - - 5 Solution A suitable denominator is ( )( 5) - - ( ) ( - 5) - 5 ( -) ( - 5) Epand the brackets on the numerator -4-5 4 - ( -) ( - 5) Simplify - 7 ( -) ( - 5) Eample 7 Epress 6 as a single fraction in its simplest form ( ) - ( )( - ) Solution A suitable denominator is ( )( )( ) 6 ( ) - ( )( - ) ( -) - 6( ) ( -) ( ) ( ) Epand the brackets in the numerator Simplify Factorise Cancel --6-8 - ( )( ) -7-8 - ( )( ) ( - 9) ( ) ( -) ( ) ( ) - 9 - ( ) Eercise B (answers p 9) Epress each of these as a single fraction in its simplest form (a) (b) - 4 (c) (d) - - - 5 Epress each of these as a single fraction in its simplest form (a) 5 9 (b) (c) 5 (d) - 4 - Epress each of these as a single fraction (a) (b) - (c) a a (d) b (e) - a b y b c a b a a 4 Epress - as a single fraction in its simplest form b b 0 Rational epressions
5 Epress each of these as a single fraction in its simplest form - (a) (b) (c) - 6 - - - 5 (d) (e) (f) 4 - - 5 4 - -- 6 Epress each of these as a single fraction in its simplest form (a) 6 (b) ( - 4) ( ) ( -) ( -) (c) 5 (d) 7 4-7 (e) ( ) - ( ) ab bc 4 7 (a) Factorise the denominators in the sum -- 7 6 (b) Show that this sum is equivalent to - 8 Epress each of these as a single fraction in its simplest form (a) 6 8 (b) 5 9-5 6 8 (c) 5 (d) - 9-4 - 9 Epress 5 as a single fraction in its simplest form 5 - - 5 0 A function is defined by f(), $ 0 (a) Evaluate f(5) as a single fraction in its simplest form (b) Prove that f() $ for all positive values of Functions are defined by g() ( 7 ) and h() ( 7 4) - - 4 Show that gh() - 4 7 - (a) Epress as a single fraction (b) Hence write the epression as a single fraction (f) ( - ) - ( - )( ) z - y 4yz ( )( ) - - - 7 7 Rational epressions
D C Etension: Leibniz s harmonic triangle (answers p 9) This section provides an opportunity to apply the techniques of sections A and B to some fraction patterns It also provides valuable practice in forming conjectures and proving them The method introduced to add a series by writing each term as a difference is not part of the content for Core 4 Gottfried Leibniz (646 76) was a German philosopher and mathematician who is best known for his work on calculus The distinguished Dutch physicist and mathematician Christian Huygens (69 69) challenged Leibniz to calculate the infinite sum of the reciprocals of the triangle numbers: 6 0 C Show that the nth term of this series is nn (You need to know that the nth triangle number is n(n ) ) C (a) Show that each term can be written as the difference - n n (b) Hence, show that the sum of the first n terms can be written: Rational epressions (! @) (@ -) (- $) ($ %) (% ^) (c) Hence find a formula for the sum of the reciprocals of the first n triangle numbers Write your formula as a single fraction (d) Use your formula to find the sum of the reciprocals of the first five triangle numbers Check your result by adding the appropriate fractions C Now, think about the sum to infinity of the reciprocals of the triangle numbers Show that, as n gets larger, the sum gets closer and closer to Eercise C (answers p 40) In the course of his work on summing infinite series, Leibniz devised a triangle which he called the harmonic triangle Part of this triangle is 6 4 D D 4 5 { 5 The fractions on each edge form a sequence of unit fractions where the denominators increase by each time Each fraction in the triangle is the sum of the two fractions below it Verify that the sum of and { is D Ê - ˆ Ë n n In their simplest form, find the fractions in the net row of the harmonic triangle
Consider the fractions in the diagonals of the triangle Row Row Diagonal Diagonal Diagonal Row 6 Row 4 4 D D 4 Row 5 5 { 5 Diagonal 4 Diagonal 5 What fraction appears in row 0 and diagonal? 4 (a) Show that in diagonal the kth fraction and its successor can be written as and k k (b) Show that the sum of the first n fractions in diagonal can be written as ( ) ( ) ( 4) (4 5) (5 6) ( - n n ) (c) Hence find a formula for the sum of the first n fractions in diagonal Write your formula as a single fraction (d) Use your formula to find the sum of the first four fractions in diagonal Check your result by adding the appropriate fractions (e) What will happen to the sum of the first n fractions in diagonal as n gets larger and larger? Justify your answer (f) Prove that the kth fraction in diagonal can be written as kk (g) (i) Show that the fraction appears in diagonal 40 (ii) In which row is? 40 5 (a) Show that in diagonal the kth fraction and its successor can be written as and kk ( k ) ( k ) (b) Hence find a formula for the sum of the first n fractions in diagonal (c) What will happen to the sum of the first n fractions in diagonal as n gets larger and larger? Justify your answer (d) (i) Prove that the kth fraction in diagonal can be written as kk ( ) ( k ) (ii) What is the 0th fraction in diagonal? *6 Investigate the other diagonals in the harmonic triangle Can you find an epression for the nth fraction in diagonal m? Can you find an epression for the sum of the first n fractions in diagonal m? What happens to the sum of the first n fractions in diagonal m as n gets larger? *7 Prove that the sum of the reciprocals of any pair of consecutive 4 triangle numbers T n and T n is T T - n n Rational epressions
D Etension: the harmonic mean (answers p 4) This section provides an opportunity to apply the techniques of sections A and B to a new type of average, the harmonic mean It includes some challenging work on proving statements The harmonic mean itself is not part of the content for Core 4 You will be familiar with the arithmetic mean of two numbers (half of the sum) and possibly their geometric mean (the square root of the product) The harmonic mean was probably so called because it can be used to produce a set of harmonious notes in music One of the earliest mentions of it is in a surviving fragment of the work of Archytas of Tarentum (circa 50 BCE) who was a contemporary of Plato He wrote There are three means in music: one is arithmetic, the second is geometric, and the third is the subcontrary, which they call harmonic Consider two lengths of 4 and Add half of 4 to 4 and subtract half of from 4 4 6 4 ( of 4) 6 and ( of ) 6, so we say that the harmonic mean of 4 and is 6 For any two numbers, if you find a fraction 4 Rational epressions so that smaller number ( p of smaller number ) is the same as larger number ( p q q of larger number ) then the value of these two epressions is the harmonic mean of the two numbers D (a) Write down (i) 6 (- of 6) (ii) 0 (- of 0) (b) Hence write down the harmonic mean of 6 and 0 D Try various fractions until you find the harmonic mean of (a) 6 and 8 (b) 0 and 5 (c) 4 and 8 p q 4 of 4 of D (a) If k is a fraction so that a ka b kb, find an epression for k in terms of a and b ab (b) Hence show that the harmonic mean of a and b can be epressed as a b (c) Use this rule to find the harmonic mean of 0 and 90
ab D4 (a) Show that a b a b (b) Hence show that the harmonic mean of two numbers is the reciprocal of the mean of their reciprocals ab The harmonic mean of two numbers is usually defined as or a b a b Eercise D (answers p 4) (a) In certain situations, usually where the average of two rates is needed, it is appropriate to find the harmonic mean Suppose a car travels from Harton to Monyborough at a speed of mph It makes the return journey at a speed of y mph Show that the average speed in mph for the whole return journey is the harmonic mean of and y (b) Hence find the average speed for a car that makes the return journey from Harton to Monyborough at a speed of 60 mph on the way out and 0 mph on the way back (a) Show that is the harmonic mean of and 6 (b) Show that the reciprocals of these three numbers, listed in order of size, form an arithmetic sequence (c) Show that if p, q and r are three numbers such that q is the harmonic mean of p and r, then, and r q p form an arithmetic sequence * A square of side is inscribed in a triangle so that one side lies along the base as shown Prove that is half the harmonic mean of the base of the triangle and its height (as measured from that base) *4 The diagram on the right is a trapezium A parallel line segment is drawn through the point of intersection of the two diagonals Let the lengths of the three parallel lines be a, b and c as shown Prove that b is the harmonic mean of a and c a b c Rational epressions 5
Key points When working with rational epressions factorise all epressions where possible cancel any factors common to the numerator and denominator (pp 6 0) To divide by a rational epression you can multiply by its reciprocal (p 7) To add or subtract rational epressions, write each with the same denominator (pp 9 0) Test yourself (answers p 4) - Epress as a single fraction in its simplest form ) - - 4 5 0 5 A function is defined by f() where is an integer such that ^ 0 6 5 (a) Evaluate f() in its simplest fractional form (b) Prove that f() can always be epressed as a fraction with as its numerator A function is defined by f() 5 4, $ 0 4 (a) Show that f() (b) Hence solve the inequality f() $ 4 Epress each of these as a single fraction, in its simplest form where appropriate y (a) - (b) (c) 4 - y 5 Show that 6 6 y - y y y ( - ) - - - 6 Epress as a single fraction in its simplest form - ( ) - - ( ) 7 Epress 0 as a single fraction in its simplest form ( ) 5-5 8 Epress -9 as a single fraction in its simplest form - -- 5 9 A function is defined by f(), $ - - -4 8 Show that f() 4 6 Rational epressions