Let the fantastic wealth of resources below teach you all about Proof by Contradiction.

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 **Proof by Contradiction** 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 2024. Please find below:

1. About Proof by Contradiction

2. Methods of Proof – Exam Worksheet & Theory Guides

3. Methods of Proof – Worksheets & Solutions

4. AH Maths Past Exam Worksheets by Topic

5. AH Maths Past Paper Questions by Topic

6. AH Maths Past & Practice Exam Papers

7. AH Maths 2020 Specimen Exam Paper

8. AH Maths Prelim & Final Exam Practice Papers

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

11. Text Book Recommended Timings & Questions – Unit One

12. Text Book Recommended Timings & Questions – Unit Two

13. Text Book Recommended Timings & Questions – Unit Three

14. AH Maths Practice Unit Assessments – Solutions Included

16. AH Maths Recommended Text Book

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

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**1. About Proof by Contradiction
**

To learn about Proof by Contradiction please click on any of the Theory Guide links in Section 2 below. There are also excellent worksheets on this topic in Section 3. Worksheets including actual SQA Exam Questions are highly recommended.

If you would like more help understanding **Proof by Contradiction **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.

**Proof by Contradiction**

This method assumes that the statement is false and then shows that this leads to something we know to be false (a contradiction). Therefore the original assumption must be false, which means that the statement is true.

**Example Questions**

Prove each of these conjectures by contradiction.

- √7 is irrational
- If n³ is odd, then n is odd
- If m² = 10, then m is not a rational number
- If the sum of 2 real numbers is irrational, at least 1 of the numbers is irrational

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**2. Methods of Proof – 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 | Methods of Proof (Number Theory) Exam Questions | Answers |

AH Maths Formulae List | AH Maths Fomulae List | |

Theory Guide 1 | Methods of Proof Theory Guide 1 | |

Theory Guide 2 | Methods of Proof Theory Guide 2 | |

Theory Guide 3 (HSN) | Methods of Proof HSN Summary Notes |

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**3. Methods of Proof – Worksheets & Solutions **

Thanks to the SQA and authors for making the excellent AH Maths Worksheets freely available for all to use. Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

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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**4. 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.

Number ______ | Topic ____________________________________________ | Answers _________ | Worked Solutions ___________________ |

1 | Binomial Theorem | Answers | In Online Study Pack |

2 | Complex Numbers | Answers | In Online Study Pack |

3 | Differentiation | Answers | In Online Study Pack |

4 | Differentiation (Further) | Answers | In Online Study Pack |

5 | Differential Equations - Variables Separable | Answers | In Online Study Pack |

6 | Differential Equations (Further) | Answers | In Online Study Pack |

7 | Functions & Graphs | Answers | In Online Study Pack |

8 | Integration | Answers | In Online Study Pack |

9 | Integration (Further) | Answers | In Online Study Pack |

10 | Matrices | Answers | In Online Study Pack |

11 | Number Theory - Methods of Proof | Answers | In Online Study Pack |

12 | Number Theory (Further) - Euclidean & Number Bases | Answers | In Online Study Pack |

13 | Partial Fractions | Answers | In Online Study Pack |

14 | Sequences & Series | Answers | In Online Study Pack |

15 | Sequences & Series - Maclaurin | Answers | In Online Study Pack |

16 | Systems of Equations | Answers | In Online Study Pack |

17 | Vectors | Answers | In Online Study Pack |

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**5. 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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**6. 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.

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**7. 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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**8. 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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**9. 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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**10. 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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**11. 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 _______________________ | Worked Solutions ___________________ |

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

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

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

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

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

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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 _______________________________ | Worked Solutions/Notes __________________________________________________________________________________ |

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

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

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 | In Online Study Pack |

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

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 _______________________ | Worked Solutions ___________________ |

Gaussian Elimination | Page 265 | Exercise 14.4 | Q1a,b,c,d,2a,b,c | In Online Study Pack |

Redundancy & Inconsistency | Page 268 | Exercise 14.6 | Q1a,b,c,2 | In Online Study Pack |

Redundancy SQA Question | 2016 Q4 (SQA) | In Online Study Pack | ||

Inconsistency SQA Question | 2017 Q5 (SQA) | In Online Study Pack | ||

ILL Conditioning | Page 274 | Exercise 14.9 | Q2a,b,c,d | In Online Study Pack |

ILL Conditioning SQA Question | 2012 Q14c (SQA) | In Online Study Pack |

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**Functions & Graphs**

Subtopic ______________________________ | Page Number _____________ | Exercise ___________ | Recommended Questions _______________________ | Worked Solutions ___________________ |

Sketching Modulus Function y = |x| | Page 66 | Exercise 5.2 | Q1-9 | In Online Study Pack |

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

Odd & Even Functions | Page 74 | Exercise 5.8 | Q3a-l | In Online Study Pack |

Vertical Asymptotes & Behaviour | Page 75 | Exercise 5.9 | Q1a-f | In Online Study Pack |

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

Sketching Graphs | Page 77 | Exercise 5.11 | Q1a,c,e,i,k | In Online Study Pack |

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**Differential Calculus**

Subtopic ___________________________ | Page Number ____________ | Exercise ___________ | Recommended Questions _______________________ | Worked Solutions ___________________ |

Derivative from First Principles | Page 45 | Exercise 4.1 | Q1,3,5,7 | In Online Study Pack |

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

The Product Rule | Page 51 | Exercise 4.5 | Q1a-h,Q2b,Q3a-l | In Online Study Pack |

The Quotient Rule | Page 52 | Exercise 4.6 | Q1,2,3,4 | In Online Study Pack |

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

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

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

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

Nature & Sketching Polynomials | Page 70 | Exercise 5.5 | Q1a,b,c,2a,b | In Online Study Pack |

Concavity | Page 73 | Exercise 5.7 | Q5a,b,c,Q1a,b | In Online Study Pack |

Applications | Page 187 | Ex 11.1 | Q1a,b,e,f,2a,c,3a,c | In Online Study Pack |

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**Integral Calculus**

Subtopic ______________________________________ | Page No __________ | Exercise ___________ | Recommended Questions _____________________ | Worked Solutions ___________________ |

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

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

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

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

Further Integration by Substitution | Page 105 | Exercise 7.3 | Q6a,b,c,d | In Online Study Pack |

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

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

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

Area between curve & x-axis | Page 120 | Exercise 7.10 | Q1,3 | In Online Study Pack |

Area between curve & y-axis | Page 120 | Exercise 7.10 | Q6,7 | In Online Study Pack |

Volume - revolved around x-axis SQA Question | 2014 Q10 (SQA) | In Online Study Pack | ||

Volume - revolved around y-axis SQA Question | 2017 Q16 (SQA) | In Online Study Pack | ||

Volume - revolved around x-axis | Page 120 | Exercise 7.10 | Q11,12 | In Online Study Pack |

Applications of Integral Calculus | Page 187 | Exercise 11.1 | Q4,14 | In Online Study Pack |

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**12. 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 _____________________ | Worked Solutions ___________________ |

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

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

Implicit & Explicit Functions - 1 | Page 89 | Exercise 6.4 | Q1,Q2 | In Online Study Pack |

Implicit & Explicit Functions - 2 | Page 89 | Exercise 6.4 | Q5,Q9,Q4 | In Online Study Pack |

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

Logarithmic Differentiation | Page 92 | Exercise 6.6 | Q1,Q2 | In Online Study Pack |

Parametric Equations | Page 95 | Exercise 6.7 | Q1a,b,c | In Online Study Pack |

Parametric Eqns - Differentiation | Page 96 | Exercise 6.8 | Q1,2,3 | In Online Study Pack |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(i) | In Online Study Pack |

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

Applications of Further Differentiation | Page 193 | Exercise 11.2 | Q1,Q2,Q3 | In Online Study Pack |

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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 __________________________ | Worked Solutions ___________________ |

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

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

Integration by Parts - 1 | Page 116 | Exercise 7.8 | Q1a-l | In Online Study Pack |

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

Integration by Parts - 3 | Page 116 | Exercise 7.8 | Q5a,b,Q6a,b | In Online Study Pack |

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

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

First Order Diff Eqns - General Soln | Page 128 | Exercise 8.1 | Q1a-j | In Online Study Pack |

First Order Diff Eqns - Particular Soln | Page 128 | Exercise 8.1 | Q2a-g | In Online Study Pack |

Differential Equations in Context | Page 131 | Exercise 8.2 | Q2,4,5,6 | In Online Study Pack |

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**Complex Numbers**

Subtopic _________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _________________________ | Worked Solutions ___________________ |

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

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

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

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

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

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

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

Expanding Trig Formula | Page 219 | Exercise 12.6 | Q5,6,7a | In Online Study Pack |

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

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**Sequences & Series, Sigma Notation**

Subtopic _______________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions __________________________ | Worked Solutions ___________________ |

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

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

Geometric Sequence | Page 156 | Exercise 9.3 | Q1a-e,Q2,Q3,Q5 | In Online Study Pack |

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

Finding Sum to Infinity | Page 162 | Exercise 9.5 | Q1,2,3,4,6 | In Online Study Pack |

Sigma Notation | Page 168 | Exercise 10.1 | Q1a-e,Q2a-e | In Online Study Pack |

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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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**13. 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 _________________ | Worked Solutions ___________________ |

Higher Revision On Vectors | Page 282 | Exercise 15.1 | Q6,7,8 | In Online Study Pack | |

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

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

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

Vector Equation of a Straight Line | Page 298 | Exercise 15.9 | Q2 | Lesson 3 | In Online Study Pack |

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

Angle Between 2 Planes | Page 293 | Exercise 15.6 | Q1,2,3 | Lesson 5 | In Online Study Pack |

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

Intersection of 2 Lines | Page 302 | Exercise 15.11 | Q1,2 | Lesson 7 | In Online Study Pack |

Intersection of 2 Planes using Gaussian | Page 303 | Exercise 15.12 | Q1,2 | Lesson 8 | In Online Study Pack |

Intersection of 2 Planes - Alternative | Page 303 | Exercise 15.12 | Q1,2 | In Online Study Pack | |

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

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**Matrices**

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ____________________________ | Worked Solutions ___________________ |

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

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

Properties of Matrix Multiplication | Page 236 | Exercise 13.4 | Q6a,b,7a,b,8a | In Online Study Pack |

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

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

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

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

Transformation Matrices | Page 251 | Exercise 13.10 | Q1,2,5 | In Online Study Pack |

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

Subtopic ________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _______________________ | Worked Solutions ___________________ |

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

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

Maclaurin Series - SQA Questions | SQA Questions & Answers | In Online Study Pack |

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**Further Differential Equations**

Subtopic __________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions ________________________ | Worked Solutions ___________________ |

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

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

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

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

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

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

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**Further Number Theory & Proof**

Subtopic _______________________________________ | Page Number _____________ | Exercise _________ | Recommended Questions _________________________ | Worked Solutions ___________________ |

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

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

Number Bases | Page 322 | Ex 16.5 | Q1a-d,2a-f | In Online Study Pack |

Further Number Theory - SQA Questions | SQA Questions & Answers | In Online Study Pack |

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**14. 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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**15. AH Video Links**

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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**16. ****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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**17. 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 **Proof by Contradiction** 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 2024.

Topic Links

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