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Short sets of questions on one topic. Answer them all, then check.
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Exam-style papers written to the SQA course. Answers are marked as you go and you get a grade estimate at the end.
Before you start
- Show all working on paper. In the real exam, method marks are given even when the final answer is wrong.
- Watch the clock. Roughly 1½ minutes per mark (Nat 5: 40 marks in 1 hour, then 50 marks in 1 hour 30; Higher: 55 marks in 1 hour 15, then 65 marks in 1 hour 30). Skip and come back later.
- Non-calculator paper means no calculator. Keep the calculator for Paper 2.
- Check answers by substituting back in where you can.
Paper 1
No calculator
60:00
Marks: 0
A
Grade A
Grade boundaries are approximate: A 70%, B 60%, C 50%, D 40%.
Answer review
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Quick-fire questions against the clock. Great for building speed before an exam.
60
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Formulas marked Given are on the SQA exam formula list. Learn the rest.
Algebra
Quadratic formula Given
x = (−b ± √(b² − 4ac)) / 2aDifference of squares
a² − b² = (a + b)(a − b)Completing the square
x² + bx = (x + b/2)² − (b/2)²Discriminant
b² − 4ac > 0: two roots · = 0: one root · < 0: no real rootsTrigonometry
SOH CAH TOA
sin = opp/hyp · cos = adj/hyp · tan = opp/adjSine rule Given
a/sin A = b/sin B = c/sin CCosine rule Given
a² = b² + c² − 2bc cos Acos A = (b² + c² − a²) / 2bc
Area of a triangle Given
A = ½ab sin CIdentities
sin²x + cos²x = 1 · tan x = sin x / cos xExact values
sin 30° = ½ · cos 60° = ½ · tan 45° = 1Geometry
Circle
A = πr² · C = 2πr = πdPythagoras
c² = a² + b²Volumes Given
Sphere ⁴⁄₃πr³ · Cone ⅓πr²h · Pyramid ⅓AhCylinder & prism
Cylinder πr²h · Prism A × hArc & sector
Arc = (θ/360) × πd · Sector = (θ/360) × πr²Similar shapes
Area SF = k² · Volume SF = k³Statistics
Averages
Mean = Σx / n · Median = middle · Mode = most commonStandard deviation Given
s = √(Σ(x − x̄)² / (n − 1))Quartiles
IQR = Q3 − Q1Probability
P(A) = favourable / total · P(not A) = 1 − P(A)Surds & indices
Index laws
aᵐ × aⁿ = aᵐ⁺ⁿ · aᵐ ÷ aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿSpecial indices
a⁰ = 1 · a⁻ⁿ = 1/aⁿ · a^(m/n) = (ⁿ√a)ᵐSurds
√(ab) = √a × √b · rationalise: multiply by √b/√bFunctions, graphs & vectors
Straight line
y = mx + c · y − b = m(x − a)Gradient
m = (y₂ − y₁) / (x₂ − x₁)Parabola
y = (x − a)² + b has turning point (a, b)Magnitude
|a| = √(x² + y² + z²)Higher
Circle Given
x² + y² + 2gx + 2fy + c = 0 · centre (−g, −f), r = √(g² + f² − c)Scalar product Given
a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃Addition formulae Given
sin(A ± B) = sinA cosB ± cosA sinBcos(A ± B) = cosA cosB ∓ sinA sinB
Double angle Given
sin 2A = 2 sinA cosA · cos 2A = 2cos²A − 1 = 1 − 2sin²ADifferentiation
d/dx xⁿ = nxⁿ⁻¹ · d/dx sin x = cos x · d/dx cos x = −sin xIntegration
∫xⁿ dx = xⁿ⁺¹/(n+1) + C · ∫(ax+b)ⁿ dx = (ax+b)ⁿ⁺¹/(a(n+1)) + CRecurrence limit
uₙ₊₁ = auₙ + b, −1 < a < 1 → L = b / (1 − a)Logs
log ab = log a + log b · log(a/b) = log a − log b · log aⁿ = n log a0
Angles in degrees. You can also type with your keyboard — Enter for =, Esc to clear.
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Timed challenge best scores
Mock exam history
Useful websites
SQA National 5 MathsOfficial past papers and marking instructions
BBC Bitesize — Nat 5 MathsNotes, videos and quizzes for every topic
SCHOLAR (Heriot-Watt)Online course material for Scottish qualifications
mathsrevision.com — Nat 5Notes, worksheets and worked examples
Corbett MathsVideos and practice for every maths topic
Exam tips
- Show all working. You get method marks even if the final answer is wrong.
- Paper 1 is non-calculator, Paper 2 is calculator. Know which one you're sitting.
- Read the question twice. Look for "exact value", "2 decimal places", "simplify", "factorise fully".
- Check your answer. Substitute back in — does it work?
- Sketch the triangle and label the sides before choosing SOH CAH TOA, the sine rule or the cosine rule.
- Never leave a blank. A sensible first step can still earn a mark.
Books and kit that help with Nat 5 and Higher Maths. Links go to Amazon — this site may earn a small commission at no extra cost to you. As an Amazon Associate we earn from qualifying purchases.
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Covers the National 5 and Higher Maths courses with 334 practice questions (136 Nat 5, 198 Higher) over three difficulty levels. Every question has a hint and a worked solution. Your progress is saved in this browser only — nothing is sent anywhere. Once loaded, the app also works offline.