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Higher Maths revision
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Topics
Choose a skill to begin. Each practice question is generated, checked instantly, and includes a worked solution.
Every skill on the “Skills to master” list has a practice topic.
Wave function
Answers are generated and checked by code, so every question has a verified solution. Give k as a decimal (2 d.p.) or in the form √41.
Factor theorem
(a) Show that is a factor by dividing it out. Enter the quadratic factor:
(b) Hence solve f(x) = 0. Enter every real solution, separated by commas:
Some questions have three real roots and some leave a quadratic with no real roots, as in past papers. Type negatives with a minus sign.
Trig equations
Consider the equation, for 0 ≤ x < 360:
(b) Hence solve the equation. Enter every solution in degrees to 1 d.p., separated by commas: [3 marks]
Set your calculator to degrees. Some questions contain a value that is impossible for a sine or cosine, which you must reject.
Inverse functions
Choose the correct inverse function f⁻¹(x) for:
Exponential modelling
Use natural logs (ln). Small rounding differences are accepted.
Coming soon
Practice for this skill is being built. It is on the “Skills to master” list because it has been tested in every one of the last five years.
In the meantime, try one of the topics in the study list.
This week
Your recent question activity.
Your topics
Based on your last 10 questions in each topic, weakest first.
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- Practice questions for all 17 topics, marked instantly with a full worked solution
- A personal revision plan that puts the highest-mark, weakest topics first
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Last updated 12 November 2026
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Last updated 12 November 2026
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Contact
Based on SQA Higher Maths papers, 2022 to 2026. Marks are approximate and grouped by our own topic tags.
Your study order
- Master the every-year skills. They are the safest marks on the paper.
- Build the biggest topics. Trigonometry and differentiation are worth the most.
- Don’t skip what’s been missing. A skill that is absent one year can return.
Where the marks are
Average marks a year for each topic, out of 120 (Paper 1 is 55 marks and Paper 2 is 65). Totals come from the 2022–2026 papers, grouped by our own topic tags, so treat them as approximate.
Skills to master
Skills that appeared in all 5 years, ranked by total marks.
Also in every Paper 1: a triangle diagram followed by a trig formula (about 37 marks). Area questions appear in every calculator paper.
Not seen lately
Skills missing from the 2026 papers, longest gap first. A gap doesn’t mean a skill is “due”, but these shouldn’t be skipped.
Timed mini-test
Choose a length, then work through a mixed set of generator questions. Use it to practice pace—not as a formal assessment.
Choose a test length
A timer appears while you practice. Move to another topic whenever you finish a question; your normal progress is still saved.
Mistake bank
Questions you get wrong are saved here and scheduled for another attempt.
Mock papers
A full timed paper in the exam format, covering the whole course. At the end you get your marks, an estimated grade and a breakdown by topic. These are practice papers made by this site, not official SQA papers.
Choose a format
You can move on after answering each question. Your usual marks and revision data are saved as normal.
Save or move your progress
Your progress is normally kept only in this browser. Download a backup before changing device or clearing browser data.
The backup includes your question history, mistake bank, daily activity, goals, exam date, and feedback. It does not include any personal account details.
Predicted grade
Worked out from your timed mock papers, so it reflects how you do under exam conditions.
Your current estimate
Take a timed mock paper to get your predicted grade.
How it is worked out
Your predicted grade is the average of your last three mock papers, using typical grade boundaries: A from 70%, B from 60%, C from 50% and D from 40%.
Real boundaries move a little every year, so treat it as a guide rather than a promise. Practice questions show your recent average, but only full timed papers change your grade.
Self-evaluation
Rate yourself honestly on every skill in the Higher course. Your ratings are saved and show you what to work on next.
Your revision plan
Based on your saved answers. Start with a short target, then revisit it after more practice.
Progress summary
A simple report you can share with a parent, teacher, or tutor. It contains no personal details.
Formula sheet
The formulae list printed at the front of both Higher papers, plus the ones you are expected to know yourself.
Given in the exam
You get exactly these in Paper 1 and Paper 2, so you do not need to memorise them, but you do need to know when to use them.
Circle
x² + y² + 2gx + 2fy + c = 0
represents a circle with centre (−g, −f) and radius √(g² + f² − c).
(x − a)² + (y − b)² = r²
represents a circle with centre (a, b) and radius r.
Scalar product
a.b = |a| |b| cos θ
where θ is the angle between a and b, or
a.b = a₁b₁ + a₂b₂ + a₃b₃
where a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) as column vectors.
Trigonometric formulae
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A
= 2cos²A − 1
= 1 − 2sin²A
Standard derivatives and integrals
| f(x) | f′(x) |
|---|---|
| sin ax | a cos ax |
| cos ax | −a sin ax |
| f(x) | ∫ f(x) dx |
|---|---|
| sin ax | −(1/a) cos ax + c |
| cos ax | (1/a) sin ax + c |
Not given: learn these
These are not on the formulae list, so you need to know them by heart.
Straight lines
m = (y₂ − y₁) / (x₂ − x₁)
m = tan θ
θ is the angle the line makes with the positive x-axis.
y − b = m(x − a)
m₁ × m₂ = −1
for perpendicular lines.
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Quadratics
b² − 4ac
> 0: two real roots. = 0: equal roots. < 0: no real roots. ≥ 0: real roots.
x = (−b ± √(b² − 4ac)) / 2a
Trigonometry
sin²x + cos²x = 1
tan x = sin x / cos x
a cos x + b sin x = k cos(x − α)
where k cos α = a, k sin α = b and k = √(a² + b²). Expand the form the question asks for and match coefficients.
180° = π radians
Exact values
| 30° (π/6) | 45° (π/4) | 60° (π/3) | |
|---|---|---|---|
| sin | 1/2 | 1/√2 | √3/2 |
| cos | √3/2 | 1/√2 | 1/2 |
| tan | 1/√3 | 1 | √3 |
Logs and exponentials
y = aˣ ⇔ x = logₐ y
logₐ xy = logₐ x + logₐ y
logₐ (x/y) = logₐ x − logₐ y
logₐ xⁿ = n logₐ x
logₐ a = 1, logₐ 1 = 0
Calculus
d/dx (axⁿ) = naxⁿ⁻¹
d/dx f(g(x)) = f′(g(x)) × g′(x)
the chain rule.
∫ axⁿ dx = axⁿ⁺¹/(n + 1) + c, n ≠ −1
∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / a(n + 1) + c
Area = ∫ₐᵇ (upper − lower) dx
Stationary points where f′(x) = 0.
Vectors
|a| = √(a₁² + a₂² + a₃²)
AB = b − a
Collinear: AB = k BC with a common point. Section formula: if P divides AB in the ratio m : n, then p = (n a + m b) / (m + n).
a.b = 0
means a and b are perpendicular.
Recurrence relations
uₙ₊₁ = a uₙ + b
has a limit when −1 < a < 1:
L = b / (1 − a)
Triangles
a² = b² + c² − 2bc cos A
Area = ½ ab sin C
Often used in Paper 1 with the addition formulae.
Study guides
Write exact values first, show each algebra step, and check signs before using a calculator.
Use degree mode only when asked. Keep full values in memory and round at the end.
“Hence” means use a previous result. “Show that” means make your working easy to follow.
Past-paper focus
This export contains topic evidence from the 2022–2026 papers, rather than individual paper questions. Use the finder to choose a high-value practice topic.