Optimisation

Welcome to highermathematics.co.uk

A sound understanding of Optimisation is essential to ensure exam success.

Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. A ‘good’ pass at Higher Maths will set you up well for the AH Maths Course next year should you be interested. Please do your very best to keep on top of your studies.

For students looking for extra help with the Higher Maths course in 2020/21 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

We hope the resources on this website prove useful and wish you the very best of success with your Higher Maths course in 2020/21.

Please find below:

1. Higher Maths Essential Skills 

2. Higher Maths Exam Worksheets by Topic

3. Higher Maths Past & Practice Papers by Topic 

4. Higher Maths Past & Practice Papers 

5. 40 Non-Calculator Higher Maths Questions & Answers

6. 200 Higher Maths Questions & Answers 

7. Practice Exam Papers A to H – Answers Included

8. Higher Maths Exam Check Lists 

9. About Optimisation 

10. Old Higher Maths Exam Worksheets by Topic

11. Higher Maths Text Book Solutions

12. Higher Maths Theory Guides 

13. Higher Maths Mind Maps 

14. Higher Maths Practice Unit Assessments (Solutions Included)

15. Higher Maths Past Paper Video Solutions

16. Higher Maths Recommended Text Book

17. Exam Focused Online Study Pack – For students looking for a ‘good’ Pass

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Higher Maths Resources

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1. Higher Maths Essential Skills 

Thanks to Mr Graham Rennie for sharing the excellent resources below.

The Essential Skills Worksheets can be used for general revision, homework, consolidation of a topic or preparation for assessments, tests and exams.

Essential Skills _______________	Topic __________________________	Answers ________	Essential Skills _______________	Topic _________________________	Answers ___________
Essential Skills 1	Median of a Triangle	Answers	Essential Skills 18	Logarithmic Equations	Answers
Essential Skills 2	Perpendicular Bisectors	Answers	Essential Skills 19	Proving Trigonometric Identities	Answers
Essential Skills 3	Altitude of a Triangle	Answers	Essential Skills 20	Related Graphs	Answers
Essential Skills 4	Equation of a Tangent to a Curve	Answers	Essential Skills 21	Scalar Product	Answers
Essential Skills 5	Stationary Points	Answers	Essential Skills 22	Further Differentiation	Answers
Essential Skills 6	Quadratic Inequalities	Answers	Essential Skills 23	Further Integration	Answers
Essential Skills 7	Completing the Square	Answers	Essential Skills 24	Recurrence: Consecutive Terms	Answers
Essential Skills 8	Tangent to a Circle	Answers	Essential Skills 25	Differential Equations	Answers
Essential Skills 9	Intersection Lines and Circles	Answers	Essential Skills 26	Definite Integrals	Answers
Essential Skills 10	Section Formula	Answers	Essential Skills 27	Composite Functions	Answers
Essential Skills 11	Trigonometric Formula	Answers	Essential Skills 28	Inverse Functions	Answers
Essential Skills 12	Related Angles	Answers	Essential Skills 29	Gradients and Angles	Answers
Essential Skills 13	Trigonometric Equations	Answers	Essential Skills 30	Angle between Vectors	Answers
Essential Skills 14	Synthetic Division	Answers	Essential Skills 31	Natural Logarithms	Answers
Essential Skills 15	Limit of a Recurrence Relation	Answers	Essential Skills 32	Logs: Connecting 2 Variables	Answers
Essential Skills 16	The Wave Function	Answers	Essential Skills 33	Using the Discriminant	Answers
Essential Skills 17	Derived Graphs	Answers

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2. Higher Maths 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 new CfE Higher Maths Questions below are available in the Online Study Pack.

Number ______	Topic ___________________________	Answers ____________	Number ______	Topic ___________________________	Answers _____________
1	Circles	Answers	16	Polynomials & Quadratics	Answers
2	Circles (Old Higher)	Ans Included	17	Polynomials (Old Higher)	Ans Included
3	Differentiation	Answers	18	Quadratics (Old Higher)	Ans Included
4	Differentiation (Optimisation)	Answers	19	Recurrence Relations	Answers
5	Differentiation (Old Higher)	Ans Included	20	Recurrence Relations (Old Higher)	Ans Included
6	Exponentials & Logs	Answers	21	Straight Lines	Answers
7	Exponentials & Logs (Old Higher)	Ans Included	22	Straight Lines (Old Higher)	Ans Included
8	Functions & Graphs	Answers	23	Trig Formulae & Equations	Answers
9	Functions (Old Higher)	Ans Included	24	Trig Addition Formula (Old Higher)	Ans Included
10	Further Calculus	Answers	25	Trig Graphs & Eqns (Old Higher)	Ans Included
11	Further Calculus (Old Higher)	Ans Included	26	Vectors	Answers
12	Graphs of Functions (Old Higher)	Ans Included	27	Vectors (Old Higher)	Ans Included
13	Integration	Answers	28	Wave Function	Answers
14	Integration - Area under a Curve	Answers	29	Wave Function (Old Higher)	Ans Included
15	Integration (Old Higher)	Ans Included	30	X-Mas Revision Worksheet	Solutions

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3. Higher Maths Past & Practice Papers by Topic

Thanks to the SQA for making these available.

Questions and answers have been split up by topic for ease of reference.

Worked solutions to all questions below available in the Online Study Pack.

. Paper _____________

. Marking __________

. Circles ______

Differentiation & Optimisation ____________

Exp's & Logs ________

Functions & Graphs _________

Further Calculus ________

. Integration _________

Polynomials & Quadratics ____________

Recurrence Relations __________

Straight Lines _________

Trig Formulae & Equations ____________

. Vectors ________

Wave Function _________

2019 P1

Marking P1

Q3,16

Q1

Q14

Q10,12

Q6,11,17b

Q8

Q2

Q4

Q5,7

Q13,15,17a

Q9

2019 P2

Marking P2

Q15

Q7b,11

Q9,12

Q5,8

Q2,13

Q7,10

Q4

Q1

Q6b

Q3,14

Q6a

2018 P1

Marking P1

Q4

Q6,11

Q2,11,15

Q3,14

Q10

Q7

Q1,8

Q13

Q5,9,12

2018 P2

Marking P2

Q5c,12

Q3,9

Q11

Q6

Q1

Q4,7a,10

7b,c

Q5a,b

Q2

Q8

2017 P1

Marking P1

Q2

Q8,15c

Q12

Q1,6,15a

Q3,13

Q10,15b

Q4

Q9

Q7,11

Q5

Q14

2017 P2

Marking P2

Q3,10

Q4,7

Q9

Q2

Q8

Q1

Q6,11

Q5

2016 P1

Marking P1

Q4,8

Q2,9

Q14

Q6,10,12

Q5

Q15

Q3

Q1

Q13

Q7,11

2016 P2

Marking P2

Q4

Q7

Q6

Q10,11b

3b,9

Q2,3a

Q1

Q8b,11a

Q5

Q8a

2015 P1

Marking P1

Q11,14

Q2,7

Q6

Q4,5,13

Q12,15

Q3,8

Q9

Q10

Q1

2015 P2

Marking P2

Q5

Q8

Q2

Q7a

Q4

Q3

Q1

Q7b

Q6

Q9

Specimen P1

Marking P1

Q11

Q8

Q10

Q1

Q4,7

Q2,5,9

Q6,12

Q3

Specimen P2

Marking P2

Q5

Q3b,8

Q4,7

Q9

Q3a

Q2

Q1,6

Q10

Exemplar P1

Marking P1

Q1

Q8,11

Q10

Q3

Q2,5

Q6

Q4,7

Q9

Exemplar P2

Marking P2

Q2

Q9,10

Q7

Q8

Q4

Q3

Q1

Q5,6

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4. Higher Maths Past & Practice Papers

Thanks to the SQA for making these available.

Worked solutions to all CfE Higher Papers below available in the Online Study Pack.

Year _____	Paper Type _________________	Exam Paper ______________	Marking Scheme __________________
2019	Higher (CfE)	Exam Paper	Marking Scheme
2018	Higher (CfE)	Exam Paper	Marking Scheme
2018	Specimen (CfE)	Paper 1, Paper 2	Marking Scheme
2017	Higher (CfE)	Exam Paper	Marking Scheme
2016	Higher (CfE)	Exam Paper	Marking Scheme
2015	Higher (CfE)	Exam Paper	Marking Scheme
2015	Specimen (CfE)	Specimen Paper	Marking Scheme
2015	Exemplar (CfE)	Exemplar Paper	Marking Scheme
2015	Higher (Old)	Exam Paper	Marking Scheme
2014	Higher	Exam Paper	Marking Scheme
2013	Higher	Exam Paper	Marking Scheme
2012	Higher	Exam Paper	Marking Scheme
2011	Higher	Exam Paper	Marking Scheme
2010	Higher	Exam Paper	Marking Scheme
2009	Higher	Exam Paper	Marking Scheme
2008	Higher	Exam Paper	Marking Scheme
2007	Higher	Exam Paper	Marking Scheme
2006	Higher	Exam Paper	Marking Scheme
2005	Higher	Exam Paper	Marking Scheme
2004	Higher	Exam Paper	Marking Scheme
2003	Higher	Exam Paper	Marking Scheme
2002	Higher	Exam Paper	Marking Scheme
2002	Higher (Winter Diet)	Exam Paper
2001	Higher	Exam Paper	Solutions

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5. 40 Non-Calculator Higher Maths Questions & Answers

Thanks to the SQA and author(s) for collating the questions below.

Start with these questions to help build your confidence. Once finished, move onto the 200 Higher Maths Exam Questions in the section below checking your answers as you go.

Clear, easy to follow, step-by-step worked solutions to all 40 questions below available in the Online Study Pack.

Exam Questions ______________________________________	Answers ___________________________
Whole Booklet for Printing	Answers
Sheet A - 10 Questions	Answers
Sheet B - 10 Questions	Answers
Sheet C - 10 Questions	Answers
Sheet D - 10 Questions	Answers

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6. 200 Higher Maths Questions & Answers

Thanks to the SQA and author(s) for collating the questions below.

Worked solutions to all 200 questions are available in the Online Study Pack.

Exam Questions _____________________	Answers ___________
Whole Booklet for Printing	Answers
Exam Questions 1 - 20	Answers
Exam Questions 21 - 40	Answers
Exam Questions 41 - 60	Answers
Exam Questions 61 - 80	Answers
Exam Questions 81 - 100	Answers
Exam Questions 101 - 120	Answers
Exam Questions 121 - 140	Answers
Exam Questions 141 - 160	Answers
Exam Questions 161 - 180	Answers
Exam Questions 181 - 200	Answers

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7. Practice Exam Papers A to H – Answers Included

Thanks to the SQA and Larkhall Academy for collating the excellent practice exam papers below.

Clear, easy to follow, step-by-step worked solutions to Practice Papers A to E available in the Online Study Pack.

Practice Paper _____________	Paper 1 __________	Paper 2 __________	Papers 1 & 2 ____________	Answers ___________
Paper A	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper B	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper C	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper D	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper E	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper F	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper G	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper H	Paper 1	Paper 2	Papers 1 & 2	HERE

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8. Higher Maths Exam Check Lists

Excellent check lists below to assess your Higher Maths knowledge. Please try to use these regularly for revision prior to tests, prelims and the final exam.

Description ___________________________________________________	Link ________	Acknowledgements ______________________
Higher Maths Course Description	HERE
SQA Higher Maths Exam Formulae List	HERE	Courtesy of SQA
Check List 1 - Higher Maths Formulae List Not Given in Exam	HERE
Check List 2 - Higher Maths Exam Whole Course	HERE
Check List 3 - Higher Maths Exam Trig Revision Sheet	HERE
Check List 4 - One Page Summary Guide Unit 1 (HSN)	HERE	Courtesy of HSN
Check List 5 - One Page Summary Guide Unit 2 (HSN)	HERE	Courtesy of HSN
Check List 6 - One Page Summary Guide Unit 3 (HSN)	HERE	Courtesy of HSN

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9. About Optimisation

• We need to learn to work confidently with Optimisation type questions
• The best practice is to do as many SQA past paper questions as possible
• Usually involves finding max/min areas and volumes
• First part of question is often a proof
• Second part of question requires finding one or more stationary point(s)
• To find the SP’s, set f ‘(x)= 0 and solve for x
• The nature of the SP(s) is determined using a Nature Table
• Exam questions are invariably in context and this is where past paper practice helps
• A Higher Maths question on Optimisation is shown below ⇒  solution in Online Study Pack

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Higher Maths Exam Question on Optimisation

Source: SQA 2008 Higher Exam Q6 P2

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Differentiation Resources

Thanks to Mr Graham Rennie and the SQA for sharing the excellent resources below.

Clear, easy to follow, step-by-step worked solutions to all worksheets below are available in the Online Study Pack.

Worksheets ____________________	Topic __________________________	Answers ________
Essential Skills 4	Equation of a Tangent to a Curve	Answers
Essential Skills 5	Stationary Points	Answers
Essential Skills 22	Further Differentiation	Answers
Higher Exam Questions	Differentiation	Answers
Higher Exam Questions	Differentiation (Optimisation)	Answers
Higher Exam Questions	Further Calculus	Answers

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10. Old Higher Maths Exam Worksheets by Topic

Thanks to the SQA for making these available.

Topic ________	Topic Name ___________________________	Link ________	Notes ___________________
Topic 1	Circles	HERE	Answers Included
Topic 2	Differentiation	HERE	Answers Included
Topic 3	Exponentials & Logarithms	HERE	Answers Included
Topic 4	Functions	HERE	Answers Included
Topic 5	Further Calculus	HERE	Answers Included
Topic 6	Graphs of Functions	HERE	Answers Included
Topic 7	Integration	HERE	Answers Included
Topic 8	Polynomials	HERE	Answers Included
Topic 9	Quadratics	HERE	Answers Included
Topic 10	Recurrence Relations	HERE	Answers Included
Topic 11	Straight Line	HERE	Answers Included
Topic 12	Trig Addition Formulae	HERE	Answers Included
Topic 13	Trig Graphs & Equations	HERE	Answers Included
Topic 14	Vectors	HERE	Answers Included
Topic 15	Wave Function	HERE	Answers Included

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11. Higher Maths Text Book Solutions 

Heinemann Higher Maths text book solutions below are courtesy of AHS.

These may prove extremely useful allowing students to progress their knowledge.

Topic ______________

Topic Name ____________________

Topic 1

Straight Line

1A

1B

1D

1E

1F

1G

1I

1K

1M

1N

1O

1O

Topic 2

Functions & Graphs 1

2A

2B

2C

2D

2F

2G

2I

Topic 3

Functions & Graphs 2

3A

3C

3K

3N

3O

3P

Topic 4

Recurrence

5A

5B

5B

5C

5D

5F

5H

5I

Topic 5

Differentiation

6A

6C

6D

6E

6F

6G

6H

6I

6J

6L

6M

6N

6O

6P

6Q

6R

6S

Topic 6

Polynomials

7B

7C

7D

7E

7F

7G

7H

7I

7J

Topic 7

Quadratics

8C

8D

8E

8F

8H

8I

8J

8K

Topic 8

Integration

9G

9H

9I

9L

9N

9P

9Q

Topic 9

Trigonometry

11B

11C

11D

11E

11F

11G

11H

Topic 10

Circles

12B

12D

12F

12H

12J

12K

12L

Topic 11

Vectors

13A

13C

13D

13E

13F

13G

13I

13K

13L

13M

13N

13O

13P

13Q

13R

13S

13U

Topic 12

Further Calculus

14B

14C

14C

14E

14G

14H

14I

14J

14K

Topic 13

Exp's & Logs

15C

15D

15E

15F

15G

15H

15I

15I

15J

15K

15L

Topic 14

Wave Function

16A

16C

16D

16E

16F

16G

16H

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12. Higher Maths Theory Guides 

Thanks to HSN for making these available.

Theory Guides _________________	Topic ____________________________________________	Link _______
Theory Guide 1	All Topics Unit 1 Theory (HSN)	HERE
Theory Guide 2	All Topics Unit 1 - One Page Summary Guide (HSN)	HERE
Theory Guide 3	All Topics Unit 2 Theory (HSN)	HERE
Theory Guide 4	All Topics Unit 2 - One Page Summary Guide (HSN)	HERE
Theory Guide 5	All Topics Unit 3 Theory (HSN)	HERE
Theory Guide 6	All Topics Unit 3 - One Page Summary Guide (HSN)	HERE
Theory Guide 7	All Topics Units 1,2 & 3 Theory (HSN)	HERE
Theory Guide 8	Circles Theory (HSN)	HERE
Theory Guide 9	Differentiation Theory (HSN)	HERE
Theory Guide 10	Exponentials & Logarithms Theory (HSN)	HERE
Theory Guide 11	Functions & Graphs Theory (HSN)	HERE
Theory Guide 12	Further Calculus Theory (HSN)	HERE
Theory Guide 13	Graphs Transformations Theory (Movement & Reflection)	HERE
Theory Guide 14	Graphs Transformations Summary Sheet	HERE
Theory Guide 15	Integration Theory (HSN)	HERE
Theory Guide 16	Polynomials & Quadratics Theory (HSN)	HERE
Theory Guide 17	Sequences Theory (HSN)	HERE
Theory Guide 18	Straight Line Theory (HSN)	HERE
Theory Guide 19	Trigonometry Theory (HSN)	HERE
Theory Guide 19	Vectors Theory (HSN)	HERE
Theory Guide 20	Wave Function Theory (HSN)	HERE

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13. Higher Maths Mind Maps

These will prove an excellent resource in helping you prepare for the final exam.

Mind Map ____________	Topic ____________________________	Mind Map ___________	Topic ____________________________
Mind Map 1	Circle 1	Mind Map 16	Polynomials & Quadratics
Mind Map 2	Circle 2	Mind Map 17	Polynomials
Mind Map 3	Composite Functions	Mind Map 18	Quadratics
Mind Map 4	Compound Angle Formula	Mind Map 19	Recurrence Relations 1
Mind Map 5	Differentiation (Preparation)	Mind Map 20	Recurrence Relations 2
Mind Map 6	Differentiation 1	Mind Map 21	Straight Lines 1
Mind Map 7	Differentiation 2	Mind Map 22	Straight Lines 2
Mind Map 8	Differentiation (Further)	Mind Map 23	Trigonometry 1
Mind Map 9	Functions & Graphs	Mind Map 24	Trigonometry 2
Mind Map 10	Functions	Mind Map 25	Vectors 1
Mind Map 11	Integration (Preparation)	Mind Map 26	Vectors 2
Mind Map 12	Integration 1	Mind Map 27	Vectors 3
Mind Map 13	Integration 2	Mind Map 28	Wave Function 1
Mind Map 14	Logs & Exponentials 1	Mind Map 29	Wave Function 2
Mind Map 15	Logs & Exponentials 2

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14. Higher Maths Practice Unit Assessments (Solutions Included)

Block _________	Outcome _____________________________________	Revision A ____________	Revision B _______________________
One	A1.1, A1.3, A1,4, EF1.3, RC1.3	Revision A	Revision B
One	A1.1, EF1.3, EF1.4, RC1.1, RC1.2	Practice Test 1	Practice Test 2 (No Solns)
Two	A1,2, A1.4, EF1.2, RC1.1, RC1.2, RC1.4	Revision A	Revision B
Three	A1,2, A1.4, EF1.2, RC1.1, RC1.2, RC1.4	Revision A	Revision B

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15. Higher Maths Past Paper Video Solutions

• DLB Maths

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16. Higher Maths Recommended Text Book

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17. Exam Focused Online Study Pack 

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about Optimisation to pass your final exam.

Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job.

Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack to view screenshots, examples and instructions on how to subscribe.

Please speak with your mum/dad and show them our fantastic website.

We hope the resources on this website prove useful and wish you the very best of success with your Higher Maths course in 2020/21.

Get the Study Pack - just £9.99

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