Let the fantastic wealth of resources below teach you all about vertical, horizontal and oblique asymptotes.

Covers the whole of the AH Maths course

Created by an experienced maths teacher

Resources used with students in Scottish Secondary Schools

# Asymptotes

A sound understanding of Asymptotes 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 2023. Please find below:

2. Functions & Graphs – Exam Worksheet & Theory Guides

3. Functions & Graphs – Recommended Text Book Questions

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

9. AH Maths Theory Guides

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

If you would like more help understanding Asymptotes 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.

Asymptotes

Note 1

Consider y = 1/x

Vertical asymptotes of y = 1/x

Look at the denominator

Since x cannot be zero then y is undefined

Therefore there is a vertical asymptote at x = 0

Behaviour either side of the V.A.

If y is +’ve (x > 0) there is positive infinity – approaches from right

If y is -‘ve (x < 0) there is negative infinity – approaches from left

or, use a table below and go either side of the asymptote:

y = 1/x

 x - 0.1 (left of asymptote) ____________________________ 0 (asymptote) ______________________ + 0.1 (right of asymptote) ____________________________ y -'ve value +'ve value Curve Approaches asymptote from left Approaches asymptote from right

.Non-vertical asymptotes of y = 1/x

Ask, is synthetic division required? (no – since degree of numerator is not ≥ degree of denominator)

As x ⇒ +/- ∞, y ⇒ 0

Therefore y = 0 is a horizontal asymptotes

Behaviour either side of the horizontal asymptote

Use the table below to help us understand the behaviour.

y = 1/x

 x + infinity - infinity y +'ve value -'ve infinity Curve Approaches asymptote from above Approaches asymptote from below

Sketch

Analysis of both the vertical and horizontal asymptotes above, the sketch below can be drawn:

Note 2

Given y = (x + 3)/((x + 2)

Synthetic division required (since the degree of numerator ≥ degree of denominator)

Here we divide to get:

y = 1 + 1/(x + 2)

Horizontal asymptote y = 1

Vertical asymptote x = -2

Exam Question

Source: SQA AH Maths Paper 2001 Question A8

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2. Functions & Graphs – 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 Functions & Graphs Exam Questions Answers AH Maths Formulae List AH Maths Fomulae List Theory Guide 1 Properties of Functions Theory Guide Theory Guide 2 (HSN) Functions Theory Guide (HSN)

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3. Functions & Graphs – 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 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

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.

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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.

 . 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 _________________ Specimen P1 Marking Q2 Q4 Q6 Q8 Q3 Q5 Q1 Q7 Specimen P2 Marking Q3 Q1 Q2,4,8,10 Q7 Q11 Q5 Q13 Q9 Q6 Q12 2019 Marking Q9 Q4 Q1a,b,6 Q1c,5,10 Q16b Q16a Q3 Q18 Q7,17 Q2 Q15 Q11,14 Q12 Q13 Q8 2018 Marking Q3 Q2 Q1b Q1a,c,6,13 Q8 Q15a Q16a Q4,10 Q14 Q17 Q7,11 Q16 Q9,12 Q5 Q15b 2017 Marking Q1 Q2 Q3 Q11,18 Q16 Q6 Q12 Q5 Q17 Q4,10 Q7 Q15 Q13 Q8 Q9 Q14 2016 Marking Q3 Q13 Q1a,b Q1c,11 Q13 Q9 Q12 Q4 Q8 Q2 Q6 Q7 Q14 Q5,10 Q16 Q15 2015 Marking Q1,9 Q2 Q4,6,8 Q17 Q10 Q14 Q13 Q3 Q5,11 Q15 Q12 Q7 Q18 Q16 2014 Marking Q2 14b Q1,13 Q1,4,6 Q10,12 Q15 Q11 Q3 Q16 Q14 Q9 Q7 Q5 Q7 Q8 2013 Marking Q1 Q2 Q11 Q4,6 Q8 Q13 Q7,10 Q17 Q3 Q15 Q9,12 Q5 Q16 Q14 2012 Marking Q4 15a Q1 Q12,13 Q8 Q11 Q7 Q14 Q3,16b Q2 Q6 Q9 Q5 16a Q10 Q15 2011 Marking Q2 Q1 3b,7 3a Q1,11a Q1,11,16 Q6 Q10 Q8,13 Q5 Q4 Q15 Q12 Q9 Q14 2010 Marking Q5 Q1 Q13 Q15 Q3,7 Q10 Q16 Q2 Q9 Q4,14 Q6 Q8,12 Q11 2009 Marking Q8 Q14 Q1a Q1b,11 Q5,7 Q9 Q13 16a Q6 Q12 Q14 Q2 Q16 Q4 Q10 Q3 Q15 2008 Marking Q8 Q4 Q10,15 Q2,5 Q4,9,10 Q7 Q3 Q16 Q1 Q12 Q6 Q14 Q11 Q13 2007 Marking Q1 Q4 Q2 Q13 Q4,10 Q4 Q16 Q3,11 Q9 Q6 Q5 Q15 Q12 Q7 Q14 Q8 Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed

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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

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 ___________________ 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

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 ___________________ 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

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 _____________________ 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

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 ___________________ 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

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 ___________________ 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

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 ___________________ 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

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 ________________________ 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

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 ___________________ 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)

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 ___________________ 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

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 ___________________ 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

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 ___________________ 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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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 Asymptotes 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 2023.

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• “The impact of these fantastic resources across all three maths websites is that 1000’s of students throughout Scotland, year on year, are gaining increased career and college course opportunities. 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.

I’d highly recommend the Online Study Pack to all AH Maths students looking for a little extra help to ensure exam success.

It’s great that over 90% of the material on this site is absolutely free for everyone to use! My students and I regularly access the Online Study Pack in class to compare our answers with the correct, easy to follow, laid out solutions to past and practice papers. The clear, easy to follow, worked solutions to recommended questions from the Math In Action text book are proving a huge help enabling students to better prepare for the exam.

I’m fully expecting the vast majority of my students to gain an ‘A’ Pass in Advanced Higher Maths in 2022.  The exam focused resources are tremendous value which have, quite simply, helped change the paths of thousands of students by giving them increased university course opportunities” – A happy maths teacher 🙂

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