Higher Maths exam questions by topic
Five free PDF practice books, each focused on one strand of the Scottish Higher course, from algebraic skills to integral calculus. Every book has eight modules of 45 numbered questions, with marks shown on every question and worked answers at the back.
These are not past papers. Every question is independently written for the Higher course, so you can work through these books alongside official past papers and marking instructions.
Free to download. No sign up, no email required. Independent resources aligned to Higher Mathematics (C847 76), not produced, approved, endorsed by or affiliated with Qualifications Scotland or the former SQA.
Which Higher topics can I practise?
Five books covering the whole course. Download each book free as a PDF and print what you need. Want the method taught first? Start with our Higher lesson notebooks.
Algebraic Skills
Surds and indices, factorisation, algebraic fractions, equations and inequations, quadratics and the discriminant, functions, polynomials and algebraic reasoning.
Download the Algebraic Skills bookFunctions and Graphs
Straight lines, domain and range, graph transformations, composite and inverse functions, quadratic and polynomial graphs, exponentials and logarithms.
Download the Functions bookTrigonometry, Vectors and Geometry
Addition and double-angle formulae, the wave function, trigonometric graphs and equations, vectors and the scalar product, circles, and recurrence relations.
Download the Trigonometry bookDifferential Calculus
Differentiating powers, tangents, stationary points, the chain rule, optimisation, differentiating sine and cosine, and closed intervals.
Download the Differential Calculus bookIntegral Calculus
Integrating powers, differential equations, definite integrals, areas under and between curves, integrating sine and cosine, and the reverse chain rule.
Download the Integral Calculus bookHow is each book organised?
Every module follows the same four-stage sequence of 45 questions, with marks rising as the demand builds. Worked answers are collected at the back of each book, and a scorecard helps you decide which modules to retest.
The four stages in every module
- Core Skills, questions 1 to 18: secure the basic method before the demand rises
- Course Practice, questions 19 to 30: apply the method with growing independence
- Exam Practice, questions 31 to 40: work at exam pace without notes
- Course-Standard Challenge, questions 41 to 45: finish the module and check readiness
What else is inside
- a contents and study plan page with score boxes, so weak modules get flagged for a second attempt
- a five-step routine on every book: attempt, show, mark, repair, then retest after 2 to 7 days
- an error-labelling routine, so students can tell an interpretation error from an algebra, restriction, arithmetic or conclusion error
- marks shown on every question and worked answers for every question at the back
- exact values expected throughout: surds, fractions and π stay exact unless a question asks for rounding
What do the five books cover?
The books split the Higher course into five strands. The order is our own organisation, not an official unit structure, so use the topic list on each card to find what you need.
| Book | Modules | Questions per module | Main skills |
|---|---|---|---|
| Algebraic Skills | 8 | 45 | Surds, polynomials, quadratics, functions |
| Functions and Graphs | 8 | 45 | Lines, transformations, logs and exponentials |
| Trigonometry, Vectors and Geometry | 8 | 45 | Trig formulae, wave function, vectors, circles |
| Differential Calculus | 8 | 45 | Tangents, stationary points, optimisation |
| Integral Calculus | 8 | 45 | Definite integrals, areas, differential equations |
The questions were written against the current Higher Mathematics course specification without reproducing any past-paper wording, diagrams, data or marking instructions. Course references identify curriculum alignment only, and the marks are an independent teaching allocation rather than official marks.
How should a student use these?
Work one module at a time, mark against the worked answers, label each error, and retest anything weak after 2 to 7 days. Paper 1 is non-calculator, so keep values exact. Official past papers and marking instructions are published free by Qualifications Scotland, and they work well alongside these books.
If the exam is months away
Work through the books in the order the topics are taught in class, one or two modules a week. A correction only counts as secure when you can reproduce it from a blank page a few days later.
If the exam is weeks away
Go straight to the Exam Practice and Course-Standard Challenge stages of your weakest strands. Calculus carries a lot of marks at Higher, so prioritise both calculus books if time is short.
If you are aiming for an A
Start every module at the Course-Standard Challenge to find your ceiling fast, then drop back only where marks are lost. Higher rewards a clear, justified method, not just a correct number.
Exam habit: keep answers exact
On Paper 1, leave surds, fractions and π in your answer and simplify fully. The square root of 64 is 8, and the square root of 324 is 18.
Exam habit: check by reversing
Substitute a solution back into the original equation, and differentiate an integral to check you get back the expression you started with. Both take seconds and catch slips.
Exam habit: state the interval
Write down the interval you are solving in, any excluded values, and whether you are working in degrees or radians. Many trigonometry marks are lost on the interval, not the method.
Practice book questions
The short version: the books are free for personal use, they include worked answers, and they are written for Scottish Higher Mathematics, grades A to D.
Are these real Higher past paper questions?
No. Every question is independently written for the Higher Mathematics course, without reproducing any past-paper wording, diagrams, data or marking instructions. You can use these books and official past papers together without overlap.
Do the books include answers?
Yes. Every book collects worked answers at the back, and every question shows its marks. Students also label each error, for example as interpretation, algebra, restriction, arithmetic or conclusion, so they can see what actually keeps going wrong.
Are they free? What is the licence?
The Free Website Edition is free for personal and household educational use. Tutors and schools must obtain the appropriate licence before copying for learners, and resale, paid redistribution and third-party uploading are prohibited.
How is the Higher Maths exam set out?
There are two papers. Paper 1 is non-calculator, worth 55 marks in 1 hour 15 minutes. Paper 2 allows a calculator and is worth 65 marks in 1 hour 30 minutes. A formulae list is provided, and the course is graded A to D.
Is this the same as GCSE Higher tier?
No. Scottish Higher Mathematics is a separate qualification, usually taken after National 5. It includes calculus, logarithms, vectors and the wave function, which go well beyond GCSE Higher tier.
How are these different from your Higher lesson notebooks?
They do different jobs. The lesson notebooks teach each topic with explanations, worked examples and four practice questions per chapter. These practice books assume the method has been taught and exist to build fluency and exam pace across eight modules per book.
My child keeps scoring low on the same module. What next?
Repeating the same questions rarely fixes it, because the problem is usually the method rather than the practice. If a module scores low after two honest attempts, that is the point to get the method retaught, whether by a teacher at school or a tutor. The first 30 minutes with one of our maths tutors are free.
Practice finds the gaps. A tutor closes them
Our maths tutors teach one to one on Zoom, £30 a lesson, and the first 30 minutes are free with no card details. If a module keeps scoring low after two attempts, that is usually the sign to get help with the method rather than repeat the questions.