Let the fantastic wealth of resources below teach you all about Integration by Substitution.

Covers the whole of the AH Maths course

Created by an experienced maths teacher

Resources used with students in Scottish Secondary Schools

Welcome to **advancedhighermaths.co.uk**

A sound understanding of **Integration by Substitution** is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH Maths course you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

To access a wealth of additional **free resources by topic** please either use the above Search Bar or click HERE selecting on the topic you wish to study.

We hope you find this website useful and wish you the very best of success with your AH Maths course in 2021/22. Please find below:

1. About Integration by Substitution

2. Integration Resources – Exam Worksheet & Theory Guides

3. Further Integrations Resources – Exam Worksheet & Theory Guides

4. Integration – Recommended Text Book Questions

5. Further Integration – Recommended Text Book Questions

6. AH Maths Past Exam Worksheets by Topic

7. AH Maths Past Paper Questions by Topic

8. AH Maths Past & Practice Exam Papers

9. AH Maths 2020 Specimen Exam Paper

10. AH Maths Prelim & Final Exam Practice Papers

12. AH Maths Course Outline, Formulae Sheets & Check List

13. Text Book Recommended Timings & Questions – Unit One

14. Text Book Recommended Timings & Questions – Unit Two

15. Text Book Recommended Timings & Questions – Unit Three

16. AH Maths Practice Unit Assessments – Solutions Included

18. AH Maths Recommended Text Book

19. Exam Focused Study Pack – Students looking for a ‘good’ Pass

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**1. About Integration by Substitution**

To learn about Integration by Substitution please click on any of the Theory Guide links in Sections 2 & 3 below. For students working from the Maths In Action text book the recommended questions on this topic are given in Section 4. Worksheets including actual SQA Exam Questions are highly recommended.

If you would like more help understanding **Integration by Substitution**** **there are full, easy to follow, step-by-step worked solutions to dozens of AH Maths Past & Practice exam questions on all topics in the AH Maths Online Study Pack. Also included in the Study Pack are full worked solutions to the recommended MIA text book questions. Please give yourself every opportunity for success, speak with your parents, and subscribe to the** exam focused**** **Online Study Pack today.

**Integration by Substitution**

- involves rewriting the entire integral (including the ” dx ” and any limits) in terms of another variable before integrating
- It must be noted that the answer for an indefinite integral must always be expressed in terms of the original variable.

Example

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**Exam Question**

**Source: SQA AH Maths Paper 2017 Question 6**

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**2. Integration – Exam Worksheet & Theory Guides**

Thanks to the SQA and authors for making the excellent AH Maths Worksheet & Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions in the worksheet below are available in the Online Study Pack.

Worksheet/Theory Guides ____________________ | Resource Link ________________________________ | Answers ____________ |

AH Maths Exam Questions | Integration Exam Questions | Answers |

AH Maths Formulae List | AH Maths Fomulae List | |

Theory Guide 1 | Integration Theory Guide | |

Theory Guide 2 (HSN) | Integration HSN Summary Notes |

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**3. Further Integration – Exam Worksheet & Theory Guides**

Thanks to the SQA and authors for making the excellent AH Maths Worksheet & Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions in the worksheet below are available in the Online Study Pack.

Worksheet/Theory Guides ____________________ | Resource Link ________________________________ | Answers ____________ |

AH Maths Exam Questions | Further Integration Exam Questions | Answers |

AH Maths Formulae List | AH Maths Fomulae List | |

Theory Guide 1 | Further Integration Theory Guide 1 | |

Theory Guide 2 | Further Integration Theory Guide 2 | |

Theory Guide 3 (HSN) | Further Integration HSN Summary Notes |

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**4. Integration – Recommended Text Book Questions **

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ______________________________________ | Page No __________ | Exercise ___________ | Recommended Questions _____________________ |

Integration (Higher Revision) | Page 100 | Exercise 7.1 | Q1a-i,2a-i,3a-l,4a-f |

Integration by Substitution | Page 103 | Exercise 7.2 | Q1a,c,e,g,i,k,m,o,q,s,u,w |

Integration by Substitution - Extra Revision! | Page 103 | Exercise 7.2 | Q1b,d,f,h,j,l,n,p,r,t,v,x |

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q2a,b,c,d,4a,b,c,d |

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q6a,b,c,d |

Further Int'n by Sub'n - sin^m(x), cos^n(x) | Page 105 | Exercise 7.3 | Q7a,b,c,d,e,f |

Further Integration by Substitution - logs | Page 105 | Exercise 7.3 | Q11a,b,c,d |

Substitution & Definite Integrals | Page 107 | Exercise 7.4 | Q1a,c,e,g,i,k |

Area between curve & x-axis | Page 120 | Exercise 7.10 | Q1,3 |

Area between curve & y-axis | Page 120 | Exercise 7.10 | Q6,7 |

Volume - revolved around x-axis SQA Question | 2014 Q10 (SQA) | ||

Volume - revolved around y-axis SQA Question | 2017 Q16 (SQA) | ||

Volume - revolved around x-axis | Page 120 | Exercise 7.10 | Q11,12 |

Applications of Integral Calculus | Page 187 | Exercise 11.1 | Q4,14 |

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**5. Further Integration – Recommended Text Book Questions **

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ______________________________________ | Page No __________ | Exercise _________________ | Recommended Questions __________________________ |

Integration using Inverse Trig Functions | Page 111 | Exercise 7.6 | Q1,2,3,4a,b |

Integration using Partial Fractions | Page 113 | Exercise 7.7 | Q1a,b,2a,b,3a,b,4a,b,5a,b,6a,b |

Integration by Parts - 1 | Page 116 | Exercise 7.8 | Q1a-l |

Integration by Parts - 2 | Page 116 | Exercise 7.8 | Q2a,c,d,e,f,g,h |

Integration by Parts - 3 | Page 116 | Exercise 7.8 | Q5a,b,Q6a,b |

Integration by Parts - Special Cases - 1 | Page 118 | Exercise 7.9 | Q1a,b,c,d |

Integration by Parts - Special Cases - 2 | Page 118 | Exercise 7.9 | Q2a,b,c,d,e |

First Order Diff Eqns - General Soln | Page 128 | Exercise 8.1 | Q1a-j |

First Order Diff Eqns - Particular Soln | Page 128 | Exercise 8.1 | Q2a-g |

Differential Equations in Context | Page 131 | Exercise 8.2 | Q2,4,5,6 |

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**6. AH Maths Past Paper Exam Worksheets by Topic **

Thanks to the SQA for making these available. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths Questions below are available in the Online Study Pack.

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**7. AH Maths Past Paper Questions by Topic**

Thanks to the SQA for making these available. Questions and answers have been split up by topic for your ease of reference. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack.

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**8. AH Maths Past & Practice Exam Papers **

Thanks to the SQA for making these available. Clear, easy to follow, step-by-step worked solutions to all SQA AH Maths questions below are available in the Online Study Pack.

Year ____ | Paper Type _________________ | Exam Paper ______________ | Marking Scheme _______________________________________ |

2019 | AH Specimen | Specimen | Marking Scheme |

2019 | Advanced Higher | Exam Paper | Marking Scheme |

2018 | Advanced Higher | Exam Paper | Marking Scheme |

2017 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | AH Specimen | Specimen | Marking Scheme |

2016 | AH Exemplar | Exemplar | Marking Scheme |

2015 | Advanced Higher | Exam Paper | Marking Scheme |

2014 | Advanced Higher | Exam Paper | Marking Scheme |

2013 | Advanced Higher | Exam Paper | Marking Scheme |

2012 | Advanced Higher | Exam Paper | Marking Scheme |

2011 | Advanced Higher | Exam Paper | Marking Scheme |

2010 | Advanced Higher | Exam Paper | Marking Scheme |

2009 | Advanced Higher | Exam Paper | Marking Scheme |

2008 | Advanced Higher | Exam Paper | Marking Scheme |

2007 | Advanced Higher | Exam Paper | Marking Scheme |

2006 | Advanced Higher | Exam Paper | Marking Scheme |

2005 | Advanced Higher | Exam Paper | Marking Scheme |

2004 | Advanced Higher | Exam Paper | Marking Scheme |

2003 | Advanced Higher | Exam Paper | Marking Scheme |

2002 | Advanced Higher | Exam Paper | Marking Scheme |

2001 | Advanced Higher | Exam Paper | Marking Scheme |

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**9. AH Maths 2020 Specimen Exam Paper **

Please find below two Specimen Papers courtesy of the SQA. Clear, easy to follow, step-by-step worked solutions to the SQA AH Maths Specimen Paper available in the Online Study Pack.

. Date __________ | . Paper ___________ | . Marking ______ | Binomial Theorem ________ | Partial Fractions ________ | . Differentiation ___________ | Further Differentiation ___________ | . Integration ___________ | Further Integration ____________ | Functions & Graphs ___________ | Systems of Equations ____________ | Complex Numbers __________ | Seq & Series _________ | Further Seq & Series ____________ | . Matrices _________ | . Vectors __________ | Methods of Proof __________ | Further No Theory ___________ | Differential Equations ____________ | Further Differential Eqns _________________ |

June 2019 | Specimen P1 | Marking | Q2 | Q4 | Q6 | Q8 | Q3 | Q5 | Q1 | Q7 | |||||||||

June 2019 | Specimen P2 | Marking | Q3 | Q1 | Q2,4,8,10 | Q7 | Q11 | Q5 | Q13 | Q9 | Q6 | Q12 |

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**10. AH Maths Prelim & Final Exam Practice Papers**

Thanks to the SQA and authors for making these freely available. Please use regularly for revision prior to assessments, tests and the final exam. Clear, easy to follow, step-by-step worked solutions to the first five Practice Papers below are available in the Online Study Pack.

AH Practice Exam Paper _____________________ | Marking ___________ | AH Practice Exam Paper _____________________ | Marking ___________ |

Practice Exam Paper 1 | HERE | Practice Exam Paper 5 | HERE |

Practice Exam Paper 2 | HERE | Practice Exam Paper 6 | HERE |

Practice Exam Paper 3 | HERE | Practice Exam Paper 7 | HERE |

Practice Exam Paper 4 | HERE | Practice Exam Paper 8 | HERE |

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**11. AH Maths Theory Guides **

Thanks to the authors for making the excellent AH Maths Theory Guides freely available for all to use. These will prove a fantastic resource in helping consolidate your understanding of AH Maths.

**Unit One**

Topic 1 ______________________ | Topic 2 ___________________ | Topic 3 _____________________ | Topic 4 ___________________ | Topic 5 ___________________ | Topic 6 ___________________ |

Partial Fractions 1 | Binomial 1 | Gaussian 1 | Functions 1 | Differentiation 1 | Integration 1 |

Partial Fractions 2 | Binomial 2 | Gaussian 2 | Functions (HSN) | Differentiation 2 | Integration (HSN) |

Partial Fractions (HSN) | Binomial (HSN) | Gaussian (HSN) | Differentiation (HSN) |

**Unit Two**

Topic 1 ______________________ | Topic 2 ________________________ | Topic 3 ___________________ | Topic 4 ____________________ | Topic 5 _________________________ |

Further Differentiation 1 | Further Integration 1 | Complex Numbers 1 | Sequences & Series 1 | Methods of Proof |

Further Differentiation 2 | Further Integration 2 | Complex Numbers 2 | Sequences & Series 2 | Proof by Induction |

Differentiation (HSN) | Integration (HSN) | Complex Nos (HSN) | Seq & Series (HSN) | Methods of Proof (HSN) |

**Unit Three**

Topic 1 ________________________ | Topic 2 _________________ | Topic 3 _____________________ | Topic 4 _____________________ | Topic 5 ______________________________ |

Vectors 1 | Matrices 1 | Maclaurin Series 1 | Differential Eqns 1 | Further Number Theory |

Vectors 2 | Matrices 2 | MacLaurin Series 2 | Differential Eqns 2 | |

Vectors 3 | Matrices 3 | Maclaurin Series (HSN) | Differential Eqns (HSN) | |

Vectors (HSN) | Matrices (HSN) |

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**12. AH Maths Course Outline, Formulae Sheets & Check List**

Thanks to the SQA and authors for making the excellent resources below freely available. These are fantastic check lists to assess your AH Maths knowledge. Please try to use these regularly for revision prior to tests, prelims and the final exam.

Title ____________________________________ | Link ___________ | Courtesy ___________________ |

AH Maths Course Outline & Timings | HERE | |

SQA AH Maths Exam Formulae List | HERE | Courtesy of SQA |

SQA Higher Maths Exam Formulae List | HERE | Courtesy of SQA |

SQA AH Maths Support Notes | HERE | Courtesy of SQA |

AH Maths Complete Check List | HERE |

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**13. Text Book Recommended Timings & Questions – Unit One **

Course timings, along with specific text book exercises/questions for Unit One, courtesy of Teejay Publishers can be found HERE.

**Partial Fractions**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic _______________________________ | Page Number _____________ | Exercise _____________ | Recommended Questions _______________________ | Comment ________________ |

Type One - Partial Fractions | Page 23 | Exercise 2.2 | Q1, 5, 12, 18, 19, 22, 25 | |

Type Two - Partial Fractions | Page 24 | Exercise 2.3 | Q1, 3, 5, 10, 14, 18 | |

Type Three - Partial Fractions | Page 25 | Exercise 2.4 | Q1, 5, 7, 9, 11 | |

Algebraic Long Division Worksheet | Worksheet | Worked Solutions | ||

Partial Fraction - Long Division | Page 26 | Exercise 2.5 | Q1 a, b, e, j, l |

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**Binomial Theorem**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ____________________________________ | Page Number _____________ | Exercise ___________ | Recommended Questions _______________________________ | Notes for Lesson __________________________________________________________________________________ |

Combinations nCr | Page 33 | Exercise 3.3 | Q1a,b,c,2a,b,c,4a-d,5a,b,6a,7a,b,d | |

Expanding - Lesson 1 | Page 36 | Exercise 3.4 | Q1a,b,c,2a,i,ii,iii,iv | |

Expanding - Lesson 2 | Page 36 | Exercise 3.4 | Q3a-d,4a-f | THEORY - Questions 3 & 4 |

Finding Coefficients | Page 38 | Exercise 3.5 | Q1a,b,c,4a,5a,6 | |

Approximation eg 1.05^5 = ? | Page 40 | Exercise 3.6 | Q1a,b,c,d | |

Simplifying General Term (SQA Questions) | SQA Questions & Answers | Common SQA Binomial Questions not in AH Text Book |

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**Systems of Equations**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book are shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ______________________________ | Page Number _____________ | Exercise _______________ | Recommended Questions _______________________ |

Gaussian Elimination | Page 265 | Exercise 14.4 | Q1a,b,c,d,2a,b,c |

Redundancy & Inconsistency | Page 268 | Exercise 14.6 | Q1a,b,c,2 |

Redundancy SQA Question | 2016 Q4 (SQA) | ||

Inconsistency SQA Question | 2017 Q5 (SQA) | ||

ILL Conditioning | Page 274 | Exercise 14.9 | Q2a,b,c,d |

ILL Conditioning SQA Question | 2012 Q14c (SQA) |

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**Functions & Graphs**

Subtopic ______________________________ | Page Number _____________ | Exercise ___________ | Recommended Questions _______________________ |

Sketching Modulus Function y = |x| | Page 66 | Exercise 5.2 | Q1-9 |

Inverse Functions | Page 67 | Exercise 5.3 | Q1a,c,e,g,i,2a,c,e,3 |

Odd & Even Functions | Page 74 | Exercise 5.8 | Q3a-l |

Vertical Asymptotes & Behaviour | Page 75 | Exercise 5.9 | Q1a-f |

Horizontal & Oblique Asymptotes | Page 76 | Exercise 5.10 | Q1a,b,f,g,k,l |

Sketching Graphs | Page 77 | Exercise 5.11 | Q1a,c,e,i,k |

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**Differential Calculus**

Subtopic ___________________________ | Page Number ____________ | Exercise ___________ | Recommended Questions _______________________ |

Derivative from First Principles | Page 45 | Exercise 4.1 | Q1,3,5,7 |

The Chain Rule | Page 48 | Exercise 4.3 | Q1a,d,2a,c,3b,4a,5a |

The Product Rule | Page 51 | Exercise 4.5 | Q1a-h,Q2b,Q3a-l |

The Quotient Rule | Page 52 | Exercise 4.6 | Q1,2,3,4 |

Differentiation - A Mixture! | Page 53 | Exercise 4.7 | Q1,2,3,4,5 |

Sec, Cosec & Cot | Page 55 | Exercise 4.8 | Q1a,b,2a,c,d,3a,c,e,g |

Exponential Functions | Page 58 | Exercise 4.9 | Q1a,c,e,2a,3e,4a,b,5a,e |

Logarithmic Functions | Page 58 | Exercise 4.9 | Q1k,m,o,q,s,2f,g,3a,b,c,4d,e,5d |

Nature & Sketching Polynomials | Page 70 | Exercise 5.5 | Q1a,b,c,2a,b |

Concavity | Page 73 | Exercise 5.7 | Q5a,b,c,Q1a,b |

Applications | Page 187 | Ex 11.1 | Q1a,b,e,f,2a,c,3a,c |

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**Integral Calculus**

Subtopic ______________________________________ | Page No __________ | Exercise ___________ | Recommended Questions _____________________ |

Integration (Higher Revision) | Page 100 | Exercise 7.1 | Q1a-i,2a-i,3a-l,4a-f |

Integration by Substitution | Page 103 | Exercise 7.2 | Q1a,c,e,g,i,k,m,o,q,s,u,w |

Integration by Substitution - Extra Revision! | Page 103 | Exercise 7.2 | Q1b,d,f,h,j,l,n,p,r,t,v,x |

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q2a,b,c,d,4a,b,c,d |

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q6a,b,c,d |

Further Int'n by Sub'n - sin^m(x), cos^n(x) | Page 105 | Exercise 7.3 | Q7a,b,c,d,e,f |

Further Integration by Substitution - logs | Page 105 | Exercise 7.3 | Q11a,b,c,d |

Substitution & Definite Integrals | Page 107 | Exercise 7.4 | Q1a,c,e,g,i,k |

Area between curve & x-axis | Page 120 | Exercise 7.10 | Q1,3 |

Area between curve & y-axis | Page 120 | Exercise 7.10 | Q6,7 |

Volume - revolved around x-axis SQA Question | 2014 Q10 (SQA) | ||

Volume - revolved around y-axis SQA Question | 2017 Q16 (SQA) | ||

Volume - revolved around x-axis | Page 120 | Exercise 7.10 | Q11,12 |

Applications of Integral Calculus | Page 187 | Exercise 11.1 | Q4,14 |

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**14. Text Book Recommended Timings & Questions – Unit Two **

Course timings, along with specific text book exercises/questions for Unit Two, courtesy of Teejay Publishers can be found HERE.

**Further Differentiation**

Subtopic _______________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _____________________ |

Inverse Trig Functions & Chain Rule | Page 85 | Exercise 6.2 | Q1a,b,c,Q2b,c,dQ3a,d |

Inverse Trig Fns & Product/Quotient Rules | Page 86 | Exercise 6.3 | Q2,Q3 |

Implicit & Explicit Functions - 1 | Page 89 | Exercise 6.4 | Q1,Q2 |

Implicit & Explicit Functions - 2 | Page 89 | Exercise 6.4 | Q5,Q9,Q4 |

Second Derivatives of Implicit Functions | Page 90 | Exercise 6.5 | Q1a,d,f,k(i),6 |

Logarithmic Differentiation | Page 92 | Exercise 6.6 | Q1,Q2 |

Parametric Equations | Page 95 | Exercise 6.7 | Q1a,b,c |

Parametric Eqns - Differentiation | Page 96 | Exercise 6.8 | Q1,2,3 |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(i) |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(ii),Q2,Q3 |

Applications of Further Differentiation | Page 193 | Exercise 11.2 | Q1,Q2,Q3 |

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**Further Integration**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan Text Book shown below. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

Subtopic ______________________________________ | Page No __________ | Exercise _________________ | Recommended Questions __________________________ |

Integration using Inverse Trig Functions | Page 111 | Exercise 7.6 | Q1,2,3,4a,b |

Integration using Partial Fractions | Page 113 | Exercise 7.7 | Q1a,b,2a,b,3a,b,4a,b,5a,b,6a,b |

Integration by Parts - 1 | Page 116 | Exercise 7.8 | Q1a-l |

Integration by Parts - 2 | Page 116 | Exercise 7.8 | Q2a,c,d,e,f,g,h |

Integration by Parts - 3 | Page 116 | Exercise 7.8 | Q5a,b,Q6a,b |

Integration by Parts - Special Cases - 1 | Page 118 | Exercise 7.9 | Q1a,b,c,d |

Integration by Parts - Special Cases - 2 | Page 118 | Exercise 7.9 | Q2a,b,c,d,e |

First Order Diff Eqns - General Soln | Page 128 | Exercise 8.1 | Q1a-j |

First Order Diff Eqns - Particular Soln | Page 128 | Exercise 8.1 | Q2a-g |

Differential Equations in Context | Page 131 | Exercise 8.2 | Q2,4,5,6 |

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**Complex Numbers**

Subtopic _________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _________________________ |

Arithmetic with Complex Numbers | Page 207 | Exercise 12.1 | Q1,2,3,6,7,8 |

Division & Square Roots of Complex Nos | Page 209 | Exercise 12.2 | Q1a,b,c,2c,e,3a,b,f,5a,b |

Argand Diagrams | Page 211 | Exercise 12.3 | Q3a,b,d,e,f,i,6a,b,f,7a,b,c |

Multiplying/Dividing in Polar Form | Page 215 | Exercise 12.5 | Q1a,b,f,g |

De Moivre's Theorem | Page 218 | Exercise 12.6 | Q1,2,3a,4g,h,i,j |

Polynomials & Complex Numbers | Page 224 | Exercise 12.8 | Q2a,d,3a,b,4,5,6a,b |

Loci on the Complex Plane | Page 213 | Exercise 12.4 | Q1a,b,d,f,j,3a,b,4a,b,c |

Expanding Trig Formula | Page 219 | Exercise 12.6 | Q5,6,7a |

Roots of a Complex Number | Page 222 | Exercise 12.7 | Q2a,b,c,d,e,f,1a(i) |

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**Sequences & Series, Sigma Notation**

Subtopic _______________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions __________________________ |

Arithmetic Sequences | Page 151 | Exercise 9.1 | Q1a-f,2a-f,Q3,Q4,Q6 |

Finding Sum - Arithmetic Sequence | Page 153 | Exercise 9.2 | Q1a,b,c,Q3a-d,Q4a,b,Q5a |

Geometric Sequence | Page 156 | Exercise 9.3 | Q1a-e,Q2,Q3,Q5 |

Finding Sum - Geometric Sequence | Page 159 | Exercise 9.4 | Q1a-f,Q2a-d,Q3a-d,Q4 |

Finding Sum to Infinity | Page 162 | Exercise 9.5 | Q1,2,3,4,6 |

Sigma Notation | Page 168 | Exercise 10.1 | Q1a-e,Q2a-e |

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**Number Theory & Proof**

Topic _______________________________ | Lessons __________ | Questions _________ | Typed Solutions _______________ | Handwritten Solutions ______________________ | Exam Questions - Worked Solutions in Online Study Pack ______________________________________________________ |

Direct Proof | Lesson 1 | Ex 1 & 2 | Ex 1 & 2 Handwritten Solns | 2018-Q9,2015-Q12, 2010-Q8a | |

Proof by Counterexample | Lesson 2 | Ex 3 | Ex 3 Typed Solns | Ex 3 Handwritten Solns | 2016-Q10, 2013-Q12, 2008-Q11 |

Proof by Counterexample | Ex 4 | Ex 4 Typed Solns | Ex 4 Handwritten Solns | 2016-Q10, 2013-Q12, 2008-Q11 | |

Proof by Contradiction | Lesson 3 | Ex 5 | Ex 5 Typed Solns | Ex 5 Handwritten Solns | 2010-Q12 |

Proof by Contrapositive | Lesson 4 | Ex 6 | Ex 6 Typed Solns | Ex 6 Handwritten Solns | 2017-Q13 |

Proof by Induction | Lesson 5 | Ex 7 | Ex 7 Typed Solns | Ex 7 Handwritten Solns | 2014-Q7,2013-Q9,2012-Q16a,2011-Q12,2010-Q8b,2009-Q4,2007-Q12 |

Proof by Induction - Sigma Notation | Lesson 6 | Ex 8 | Ex 8 Typed Solns | Ex 8 Handwritten Solns | 2018-Q12,2016-Q5, 2013-Q9,2009-Q4 |

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**15. Text Book Recommended Timings & Questions – Unit Three **

Course timings, along with specific text book exercises/questions for Unit Three, courtesy of Teejay Publishers can be found HERE.

**Vectors**

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ________________________ | Lesson/Notes _________________ |

Higher Revision On Vectors | Page 282 | Exercise 15.1 | Q6,7,8 | |

The Vector Product - 1 | Page 286 | Exercise 15.3 | Q1,2a,b,5,7,8a,b,10 | Lesson 1 |

The Vector Product - 2 | Page 286 | Exercise 15.3 | Q3,4,6,12 | Lesson 2 |

The Equations of a Line | Page 298 | Exercise 15.8 | Q1a,b,2a,3a,c,e,5 | Lesson 3 |

Vector Equation of a Straight Line | Page 298 | Exercise 15.9 | Q2 | Lesson 3 |

The Equation of a Plane | Page 291 | Exercise 15.5 | Q1a,b,c,d,2a,b,3,4a,c,9,10 | Lesson 4 |

Angle Between 2 Planes | Page 293 | Exercise 15.6 | Q1,2,3 | Lesson 5 |

Intersection of Line & Plane | Page 300 | Exercise 15.10 | Q1a,b,c,2a,b,3,4a | Lesson 6 |

Intersection of 2 Lines | Page 302 | Exercise 15.11 | Q1,2 | Lesson 7 |

Intersection of 2 Planes using Gaussian | Page 303 | Exercise 15.12 | Q1,2 | Lesson 8 |

Intersection of 2 Planes - Alternative | Page 303 | Exercise 15.12 | Q1,2 | |

Intersection of 3 Planes | Page 307 | Exercise 15.3 | Q1a,c,2a,c | Lesson 9 |

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**Matrices**

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ____________________________ |

Basic Properties & Operations of Matrices | Page 231 | Exercise 13.1 | Q1,2,3a,4a,c,e,i,p,t,7a,f,9,10 |

Matrix Multiplication | Page 235 | Exercise 13.3 | Q1a,c,2a,c,k,m,o,3a,4,5a,c |

Properties of Matrix Multiplication | Page 236 | Exercise 13.4 | Q6a,b,7a,b,8a |

Determinant of a 2 x 2 Matrix | Page 240 | Exercise 13.6 | Q1a,b,d,h |

Determinant of a 3 x 3 Matrix | Page 247 | Exercise 13.9 | Q4a,b,c,d,5a,b |

Inverse of a 2 x 2 Matrix | Page 243 | Exercise 13.7 | Q1,2,4,8,9a,b,c |

Inverse of a 3 x 3 Matrix | Page 275 | Exercise 14.10 | Q1a,b,c,d |

Transformation Matrices | Page 251 | Exercise 13.10 | Q1,2,5 |

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**Further Sequences & Series (Maclaurin Series)**

Subtopic ________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _______________________ |

Maclaurin Series for f(x) | Page 179 | Exercise 10.5 | Q1a,b,c,d,3a,b |

Maclaurin Series - Composite Functions | Page 182 | Exercise 10.7 | Q1a,f,2a,3a,6a,7a,8a,b |

Maclaurin Series - SQA Questions | SQA Questions & Answers |

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**Further Differential Equations**

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ________________________ |

1st Order Linear Differential Equations | Page 136 | Exercise 8.3 | Q1a,b,2a,3a,b |

2nd Order Differential Equations (Roots Real & Distinct) | Page 140 | Exercise 8.4 | Q1a,b,c,2a,b |

2nd Order Differential Equations (Roots Real & Coincident) | Page 141 | Exercise 8.5 | Q1a,b,c,2a,b |

2nd Order Differential Equations (Roots Not Real) | Page 142 | Exercise 8.6 | Q1a,b,c,2a,b |

Non-Homogeneous Differential Equations (Finding General Solution) | Page 146 | Exercise 8.9 | Q1a,b,c |

Non-Homogeneous Differential Equations (Finding Particular Solution) | Page 146 | Exercise 8.9 | Q2a,b,c |

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**Further Number Theory & Proof**

Subtopic _______________________________________ | Page Number _____________ | Exercise _________ | Recommended Questions ____________________________ |

Finding the Greatest Common Divisor (GCD) | Page 318 | Ex 16.3 | Q1a,c,e,g,i |

Expressing GCD in the form xa + yb = d | Page 320 | Ex 16.4 | Q1,2,3,4 |

Number Bases | Page 322 | Ex 16.5 | Q1a-d,2a-f |

Further Number Theory - SQA Questions | SQA Questions & Answers |

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**16. AH Maths Practice Unit Assessments – Solutions Included**

Thanks to maths777 for making the excellent resources freely available for all to use. This will prove a fantastic resource in helping you prepare for assessments, tests and the final exam.

Methods in Algebra & Calculus __________________________ | Applications of Algebra & Calculus ____________________________ | Geometry, Proof & Systems of Equations ____________________________________ |

Practice 1 | Practice 1 | Practice 1 |

Practice 2 | Practice 2 | Practice 2 |

Practice 3 | Practice 3 | Practice 3 |

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**17. AH Maths Video Solutions**

Please click DLB Maths to view AH Maths Past Paper video solutions. There are also many videos showing worked examples by topic on the St Andrews StAnd Maths YouTube Channel link. Both video links are excellent resources in helping you prepare for assessments, tests and the final exam.

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**18. ****AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan **

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.

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**19. Advanced Higher Maths Online Study Pack **

Through step-by-step worked solutions to exam questions and recommended MIA text book questions available in the Online Study Pack we cover everything you need to know about **Integration by Substitution** to pass your final exam.

For students looking for a ‘good’ pass at AH Maths you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack. Subscribing could end up being one of your best ever investments.

Please give yourself every opportunity for success, speak with your parents, and subscribe to the** exam focused**** **Online Study Pack today.

We hope the resources on this website prove useful and wish you the very best of success with your AH Maths course in 2022.

Topic Links

Free resources to dozens of AH Maths topics are available by clicking on any of the links to the right.