SMP AS/A2 Mathematics. Core 4. for AQA. The School Mathematics Project
|
|
- Austin Stephens
- 5 years ago
- Views:
Transcription
1 SMP AS/A Mathematics Core 4 for AQA The School Mathematics Project
2 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/ 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 paperback 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
3 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
4 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
5 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 Solution Factorise Divide numerator and denominator by ( 4) Rational epressions
6 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 ) n n 5n n - 9 n 5n- n ( 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) (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
7 Simplify each of these (a) (b) (c) (d) (e) 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) ) ) 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() ( ) ) ) n - 6n 5 n -n-0 n n - n 9n - 5n-4 n - 9n 8 Rational epressions
8 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 Solution A suitable denominator is 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 ( ) 8 7 Rational epressions 9
9 Eample 6 Epress as a single fraction in its simplest form Solution A suitable denominator is ( )( 5) - - ( ) ( - 5) - 5 ( -) ( - 5) Epand the brackets on the numerator ( -) ( - 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 ( )( ) ( )( ) ( - 9) ( ) ( -) ( ) ( ) ( ) Eercise B (answers p 9) Epress each of these as a single fraction in its simplest form (a) (b) - 4 (c) (d) Epress each of these as a single fraction in its simplest form (a) 5 9 (b) (c) 5 (d) 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
10 5 Epress each of these as a single fraction in its simplest form - (a) (b) (c) (d) (e) (f) Epress each of these as a single fraction in its simplest form (a) 6 (b) ( - 4) ( ) ( -) ( -) (c) 5 (d) (e) ( ) - ( ) ab bc 4 7 (a) Factorise the denominators in the sum (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) (c) 5 (d) Epress 5 as a single fraction in its simplest form 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) Show that gh() (a) Epress as a single fraction (b) Hence write the epression as a single fraction (f) ( - ) - ( - )( ) z - y 4yz ( )( ) Rational epressions
11 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
12 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
13 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 ( 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
14 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
15 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 ) A function is defined by f() where is an integer such that ^ (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 ( - ) Epress as a single fraction in its simplest form - ( ) - - ( ) 7 Epress 0 as a single fraction in its simplest form ( ) Epress -9 as a single fraction in its simplest form A function is defined by f(), $ Show that f() 4 6 Rational epressions
SMP AS/A2 Mathematics AQA. Core 1. The School Mathematics Project
SMP AS/A2 Mathematics AQA Core 1 The School Mathematics Project SMP AS/A2 Mathematics writing team David Cassell, Spencer Instone, John Ling, Paul Scruton, Susan Shilton, Heather West SMP design and administration
More informationCambridge IGCSE and O Level Additional Mathematics Coursebook
Cambridge IGCSE and O Level Additional Mathematics Coursebook Second edition University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown
More informationFurther factorising, simplifying, completing the square and algebraic proof
Further factorising, simplifying, completing the square and algebraic proof 8 CHAPTER 8. Further factorising Quadratic epressions of the form b c were factorised in Section 8. by finding two numbers whose
More informationPure Core 2. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More informationAdvanced Higher Mathematics Course Assessment Specification
Advanced Higher Mathematics Course Assessment Specification Valid from August 015 This edition: April 013, version 1.0 This specification may be reproduced in whole or in part for educational purposes
More informationInternational Examinations. Advanced Level Mathematics Pure Mathematics 1 Hugh Neill and Douglas Quadling
International Eaminations Advanced Level Mathematics Pure Mathematics Hugh Neill and Douglas Quadling PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street,
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More informationESSENTIAL SAMPLE. Mathematical Methods 1&2CAS MICHAEL EVANS KAY LIPSON DOUG WALLACE
ESSENTIAL Mathematical Methods 1&2CAS MICHAEL EVANS KAY LIPSON DOUG WALLACE TI-Nspire and Casio ClassPad material prepared in collaboration with Jan Honnens David Hibbard i CAMBRIDGE UNIVERSITY PRESS Cambridge,
More informationMathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman
03 04 Mathematics syllabus for Grade and For Bilingual Schools in the Sultanate of Oman Prepared By: A Stevens (Qurum Private School) M Katira (Qurum Private School) M Hawthorn (Al Sahwa Schools) In Conjunction
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationTable of Contents. Unit 3: Rational and Radical Relationships. Answer Key...AK-1. Introduction... v
These materials may not be reproduced for any purpose. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored,
More informationAQA Level 2 Further mathematics Further algebra. Section 4: Proof and sequences
AQA Level 2 Further mathematics Further algebra Section 4: Proof and sequences Notes and Examples These notes contain subsections on Algebraic proof Sequences The limit of a sequence Algebraic proof Proof
More informationPURE MATHEMATICS Unit 1
PURE MATHEMATICS Unit 1 FOR CAPE EXAMINATIONS DIPCHAND BAHALL CAPE is a registered trade mark of the Caribbean Examinations Council (CXC). Pure Mathematics for CAPE Examinations Unit 1 is an independent
More informationMEI Core 2. Sequences and series. Section 1: Definitions and Notation
Notes and Eamples MEI Core Sequences and series Section : Definitions and Notation In this section you will learn definitions and notation involving sequences and series, and some different ways in which
More informationTEACHER NOTES FOR YEAR 12 SPECIALIST MATHEMATICS
TEACHER NOTES FOR YEAR 12 SPECIALIST MATHEMATICS 21 November 2016 CHAPTER 1: MATHEMATICAL INDUCTION A The process of induction Topic 1 B The principle of mathematical Sub-topic 1.1 induction Induction
More informationBrief Revision Notes and Strategies
Brief Revision Notes and Strategies Straight Line Distance Formula d = ( ) + ( y y ) d is distance between A(, y ) and B(, y ) Mid-point formula +, y + M y M is midpoint of A(, y ) and B(, y ) y y Equation
More informationAnswers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)
Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +
More informationPre-Calculus and Trigonometry Capacity Matrix
Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationTest Codes : MIA (Objective Type) and MIB (Short Answer Type) 2007
Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 007 Questions will be set on the following and related topics. Algebra: Sets, operations on sets. Prime numbers, factorisation of integers
More informationPearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0)
Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) First teaching from September 2017 First certification from June 2018 2
More informationA summary of factoring methods
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What
More informationMaths A Level Summer Assignment & Transition Work
Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationMathematics Revision Guides Partial Fractions Page 1 of 23 Author: Mark Kudlowski. AQA : C4 Edexcel: C4 OCR: C4 OCR MEI: C4 PARTIAL FRACTIONS
Mathematics Revision Guides Partial Fractions Page 1 of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C4 Edecel: C4 OCR: C4 OCR MEI: C4 PARTIAL FRACTIONS Version : Date: 1-04-01
More informationHigher Tier - Algebra revision
Higher Tier - Algebra revision Contents: Indices Epanding single brackets Epanding double brackets Substitution Solving equations Solving equations from angle probs Finding nth term of a sequence Simultaneous
More informationMathematics Specialist Units 3 & 4 Program 2018
Mathematics Specialist Units 3 & 4 Program 018 Week Content Assessments Complex numbers Cartesian Forms Term 1 3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z Week 1 3.1. review
More informationIB Mathematics Standard Level Revision guide
IB Mathematics Standard Level Revision guide F.G. Groeneveld TopClassTutors.ORG Copyright 2016 by F. Groeneveld All rights reserved. No part of this publication may be reproduced, distributed, or transmitted
More informationAS Maths for Maths Pack
Student Teacher AS Maths for Maths Pack September 0 City and Islington Sith Form College Mathematics Department www.candimaths.uk CONTENTS WS Numbers [Directed Numbers WS Numbers [Indices, powers WS Numbers
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
National Quali cations AHEXEMPLAR PAPER ONLY EP/AH/0 Mathematics Date Not applicable Duration hours Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions
More informationComposition of and the Transformation of Functions
1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationZETA MATHS. Higher Mathematics Revision Checklist
ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions
More informationAlgebraic. techniques1
techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them
More informationReport on the Examination
Version 1.0 General Certificate of Education (A-level) January 01 Mathematics MPC4 (Specification 660) Pure Core 4 Report on the Examination Further copies of this Report on the Examination are available
More informationTHE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied.
THE DISTRIBUTIVE LAW ( ) When an equation of the form a b c is epanded, every term inside the bracket is multiplied by the number or pronumeral (letter), and the sign that is located outside the brackets.
More informationJUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES DIRECTORATE TERM
JUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES 10 1 DIRECTORATE TERM 1 017 This document has been compiled by the FET Mathematics Subject Advisors together with Lead Teachers.
More informationSection 6.2 Long Division of Polynomials
Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to
More informationCore Mathematics 2 Unit C2 AS
Core Mathematics 2 Unit C2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C2.1 Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences
More informationMATHEMATICS. Higher 2 (Syllabus 9740)
MATHEMATICS Higher (Syllabus 9740) CONTENTS Page AIMS ASSESSMENT OBJECTIVES (AO) USE OF GRAPHING CALCULATOR (GC) 3 LIST OF FORMULAE 3 INTEGRATION AND APPLICATION 3 SCHEME OF EXAMINATION PAPERS 3 CONTENT
More informationLecture 5: Finding limits analytically Simple indeterminate forms
Lecture 5: Finding its analytically Simple indeterminate forms Objectives: (5.) Use algebraic techniques to resolve 0/0 indeterminate forms. (5.) Use the squeeze theorem to evaluate its. (5.3) Use trigonometric
More informationGottfried Wilhelm Leibniz made many contributions to modern day Calculus, but
Nikki Icard & Andy Hodges Leibniz's Harmonic Triangle MAT 5930 June 28, 202 Gottfried Wilhelm Leibniz made many contributions to modern day Calculus, but none of his contributions may have been as important
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationCourse. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.
Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationPURE MATHEMATICS AM 27
AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationTime: 1 hour 30 minutes
Paper Reference(s) 6663/0 Edecel GCE Core Mathematics C Silver Level S Time: hour 30 minutes Materials required for eamination Mathematical Formulae (Green) Items included with question papers Nil Candidates
More informationKing s Year 12 Medium Term Plan for LC1- A-Level Mathematics
King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives
More information2010 HSC NOTES FROM THE MARKING CENTRE MATHEMATICS EXTENSION 1
Contents 2010 HSC NOTES FROM THE MARKING CENTRE MATHEMATICS EXTENSION 1 Introduction... 1 Question 1... 1 Question 2... 2 Question 3... 3 Question 4... 4 Question 5... 5 Question 6... 5 Question 7... 6
More informationCenterville High School Curriculum Mapping Algebra II 1 st Nine Weeks
Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Chapter/ Lesson Common Core Standard(s) 1-1 SMP1 1. How do you use a number line to graph and order real numbers? 2. How do you identify
More informationYEAR 12 - Mathematics Pure (C1) Term 1 plan
Week YEAR 12 - Mathematics Pure (C1) Term 1 plan 2016-2017 1-2 Algebra Laws of indices for all rational exponents. Use and manipulation of surds. Quadratic functions and their graphs. The discriminant
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationWellington College Mathematics Department. Sixth Form Kick Start
Wellington College Mathematics Department Sith Form Kick Start Wellington College Mathematics Department Sith Form Kick Start Introduction There is a big step up from IGCSE to AS-Level or IB: questions
More informationa b = a a a and that has been used here. ( )
Review Eercise ( i j+ k) ( i+ j k) i j k = = i j+ k (( ) ( ) ) (( ) ( ) ) (( ) ( ) ) = i j+ k = ( ) i ( ( )) j+ ( ) k = j k Hence ( ) ( i j+ k) ( i+ j k) = ( ) + ( ) = 8 = Formulae for finding the vector
More informationFunctions, Graphs, Equations and Inequalities
CAEM DPP Learning Outcomes per Module Module Functions, Graphs, Equations and Inequalities Learning Outcomes 1. Functions, inverse functions and composite functions 1.1. concepts of function, domain and
More informationGlossary Common Core Curriculum Maps Math/Grade 9 Grade 12
Glossary Common Core Curriculum Maps Math/Grade 9 Grade 12 Grade 9 Grade 12 AA similarity Angle-angle similarity. When twotriangles have corresponding angles that are congruent, the triangles are similar.
More information(iii) converting between scalar product and parametric forms. (ii) vector perpendicular to two given (3D) vectors
Vector Theory (15/3/2014) www.alevelmathsng.co.uk Contents (1) Equation of a line (i) parametric form (ii) relation to Cartesian form (iii) vector product form (2) Equation of a plane (i) scalar product
More information8 Differential Calculus 1 Introduction
8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find
More informationBASIC MATHEMATICS. Lecture Notes & Tutorials UNIVERSITY OF NIZWA FOUNDATION INSTITUTE. Lecture Notes & Tutorials 1 MATH 001
BASIC MATHEMATICS Lecture Notes & Tutorials UNIVERSITY OF NIZWA FOUNDATION INSTITUTE Lecture Notes & Tutorials MATH 00 BASIC MATHEMATICS Lecture notes & tutorials Prepared By: The team of Mathematics instructors
More informationA BRIEF REVIEW OF ALGEBRA AND TRIGONOMETRY
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETR Some Key Concepts:. The slope and the equation of a straight line. Functions and functional notation. The average rate of change of a function and the DIFFERENCE-
More informationContents. CHAPTER P Prerequisites 1. CHAPTER 1 Functions and Graphs 69. P.1 Real Numbers 1. P.2 Cartesian Coordinate System 14
CHAPTER P Prerequisites 1 P.1 Real Numbers 1 Representing Real Numbers ~ Order and Interval Notation ~ Basic Properties of Algebra ~ Integer Exponents ~ Scientific Notation P.2 Cartesian Coordinate System
More informationWritten as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year
Written as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year 2012-2013 Basic MATHEMATICS First Year Diploma Semester - I First
More informationTrigonometry Self-study: Reading: Red Bostock and Chandler p , p , p
Trigonometry Self-study: Reading: Red Bostock Chler p137-151, p157-234, p244-254 Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant,
More informationARE YOU READY FOR CALCULUS?? Name: Date: Period:
ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*)
More informationThe Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA
Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More information6664/01 Edexcel GCE Core Mathematics C2 Bronze Level B3
Paper Reference(s) 666/01 Edecel GCE Core Mathematics C Bronze Level B Time: 1 hour 0 minutes Materials required for eamination papers Mathematical Formulae (Green) Items included with question Nil Candidates
More informationNEW SYLLABUS. 7th EDITION MATHEMATICS TEACHER S RESOURCE BOOK
7th EDITION NEW SYLLABUS MATHEMATICS TEACHER S RESOURCE BOOK CONTENTS Syllabus Matching Grid.... Scheme of Work....7 Chapter : Direct and Inverse Proportions Teaching Notes....9 Worked Solutions.... Chapter
More informationOCR 06 Algebra (Higher)
OCR 06 Algebra (Higher) 1. Simplify. Simplify 5 5 4. 1 3 8y y. 3. A function is given by y 5 3. Write an epression for the inverse of this function. 4. A value,, is input into this function. y The output,
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationScope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A)
Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Updated 06/05/16 http://www.haesemathematics.com.au/ Note: Exercises in red text indicate material in the 10A textbook
More informationAQA Level 2 Certificate in Further Mathematics. Worksheets - Teacher Booklet
AQA Level Certificate in Further Mathematics Worksheets - Teacher Booklet Level Specification Level Certificate in Further Mathematics 860 Worksheets - Teacher Booklet Our specification is published on
More informationSophomore Year: Algebra II Textbook: Algebra II, Common Core Edition Larson, Boswell, Kanold, Stiff Holt McDougal 2012
Sophomore Year: Algebra II Tetbook: Algebra II, Common Core Edition Larson, Boswell, Kanold, Stiff Holt McDougal 2012 Course Description: The purpose of this course is to give students a strong foundation
More informationName: Index Number: Class: CATHOLIC HIGH SCHOOL Preliminary Examination 3 Secondary 4
Name: Inde Number: Class: CATHOLIC HIGH SCHOOL Preliminary Eamination 3 Secondary 4 ADDITIONAL MATHEMATICS 4047/1 READ THESE INSTRUCTIONS FIRST Write your name, register number and class on all the work
More informationOCR A2 Level Mathematics Core Mathematics Scheme of Work
OCR A Level Mathematics Core Mathematics Scheme of Work Examination in June of Year 13 The Solomen press worksheets are an excellent resource and incorporated into the SOW NUMERICAL METHODS (6 ) (Solomen
More informationAlgebraic Functions, Equations and Inequalities
Algebraic Functions, Equations and Inequalities Assessment statements.1 Odd and even functions (also see Chapter 7)..4 The rational function a c + b and its graph. + d.5 Polynomial functions. The factor
More informationCalculus and Vectors, Grade 12
Calculus and Vectors, Grade University Preparation MCV4U This course builds on students previous eperience with functions and their developing understanding of rates of change. Students will solve problems
More informationHelpful Concepts for MTH 261 Final. What are the general strategies for determining the domain of a function?
Helpful Concepts for MTH 261 Final What are the general strategies for determining the domain of a function? How do we use the graph of a function to determine its range? How many graphs of basic functions
More informationMapping Australian Curriculum (AC) Mathematics and VELS Mathematics. Australian Curriculum (AC) Year 9 Year 10/10A
Mapping Australian Curriculum (AC) Mathematics and VELS Mathematics In the following document, the left hand column shows AC content that matches VELS content at the corresponding levels. Teaching programs
More informationPre-Calculus and Trigonometry Capacity Matrix
Information Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving
More informationax 2 + bx + c = 0 where
Chapter P Prerequisites Section P.1 Real Numbers Real numbers The set of numbers formed by joining the set of rational numbers and the set of irrational numbers. Real number line A line used to graphically
More informationThe Not-Formula Book for C1
Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationYear 12 Maths C1-C2-S1 2016/2017
Half Term 1 5 th September 12 th September 19 th September 26 th September 3 rd October 10 th October 17 th October Basic algebra and Laws of indices Factorising expressions Manipulating surds and rationalising
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
More informationIntroduction to Calculus
8 Introduction to Calculus TERMINOLOGY Composite function: A function of a function. One function, f (), is a composite of one function to another function, for eample g() Continuity: Describing a line
More informationSolutions to Problem Sheet for Week 11
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week MATH9: Differential Calculus (Advanced) Semester, 7 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/
More informationMaths Department. A Level Induction Booklet
Maths Department A Level Induction Booklet CONTENTS Chapter 1 Removing brackets page Chapter Linear equations 4 Chapter 3 Simultaneous equations 8 Chapter 4 Factors 10 Chapter 5 Change the subject of the
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More information2. Which of the following expressions represents the product of four less than three times x and two more than x?
Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are: basic calculators, scientific calculators, and graphing calculators up through the
More informationFundamentals. Introduction. 1.1 Sets, inequalities, absolute value and properties of real numbers
Introduction This first chapter reviews some of the presumed knowledge for the course that is, mathematical knowledge that you must be familiar with before delving fully into the Mathematics Higher Level
More informationCore Mathematics C12
Write your name here Surname Other names Core Mathematics C12 SWANASH A Practice Paper Time: 2 hours 30 minutes Paper - E Year: 2017-2018 The formulae that you may need to answer some questions are found
More informationA-LEVEL MATHS Bridging Work 2017
A-LEVEL MATHS Bridging Work 017 Name: Firstly, CONGRATULATIONS for choosing the best A-Level subject there is. A-Level Maths at Wales is not only interesting and enjoyable but is highly regarded by colleges,
More information2014 Mathematics. Advanced Higher. Finalised Marking Instructions
0 Mathematics Advanced Higher Finalised ing Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial
More informationYou must have: Ruler graduated in centimetres and millimetres, pair of compasses, pen, HB pencil, eraser.
Write your name here Surname Other names Pearson Edexcel Award Algebra Level 3 Calculator NOT allowed Centre Number Candidate Number Monday 8 May 017 Morning Time: hours Paper Reference AAL30/01 You must
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More informationAlgebra and Trigonometry
Algebra and Trigonometry 978-1-63545-098-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Jay Abramson, Arizona State
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More information