📐 A-Math Formula Sheet Breakdown: When, How & What to Use for Exams

📐 A-Math Formula Sheet Breakdown: When, How & What to Use for Exams

💬 Introduction

Struggling to revise for A-Math without drowning in formulas? 😩
Good news — we’ve got you covered! Our A-Math Formula Sheet is a super-condensed, exam-focused revision tool designed to help you revise every chapter in 10 minutes or less. 🧠⏱️

This blog is your complete guide to understanding how to use each formula effectively, when to apply them during exams, and what tricks help you score faster and smarter.

📘 Bonus: You can download the full free formula sheet directly from Sophia Education.

Let’s jump into each topic and break it all down. 💥

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📘 1. Simultaneous Equations: Solve Like a Pro

When to Use:
When you see two equations with two unknowns (x and y), often one linear and one quadratic.

How to Use:

• Isolate one variable (usually x or y).

• Substitute into the second equation.

• Always check both solutions at the end.

Formula Highlight:
Linear:
2x + 3y = 7
Quadratic + Substitution:
2y – x² = xy, then substitute y = x + 3

🎯 Exam Tip:
Always plug back your solutions to check for invalid roots, especially when logarithms or square roots are involved!

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🧮 2. Surds: Clean and Rationalize

When to Use:
When dealing with square roots that can’t be simplified fully.

How to Use:

• Use conjugate surds to rationalize denominators.

• Multiply numerator and denominator by the conjugate.

Formula Highlight:
(a + √b)(a - √b) = a² - b
Use this to eliminate surds in the denominator.

🎯 Exam Tip:
Don’t leave surds in the denominator — you’ll lose marks! Always rationalize.

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🔢 3. Indices & Exponential Equations

When to Use:
Whenever variables appear as powers (exponents).

How to Use:

• Apply index laws:

  • a^m * a^n = a^(m+n)

  • (a^m)^n = a^(mn)

• For exponential equations, rewrite both sides with the same base.

🎯 Exam Tip:
Can’t get same base? Take logs on both sides. That brings us to the next section…

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📈 4. Logarithms & Log Laws

When to Use:
Whenever variables appear inside logs or as powers.

How to Use:

• Basic Conversion: log_a b = c ↔ a^c = b

• Laws to Memorize:

  • log_a(xy) = log_a x + log_a y

  • log_a(x/y) = log_a x - log_a y

  • log_a(x^n) = n log_a x

🎯 Exam Tip:
Only positive numbers are allowed inside logs. Always check the domain of your answer!

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➗ 5. Quadratics & Inequalities

When to Use:
For equations in the form ax² + bx + c = 0

How to Use:

• Quadratic Formula:
  x = [-b ± √(b² - 4ac)] / 2a

• Discriminant Insight:

  • b² - 4ac > 0: 2 real roots

  • = 0: 1 real root

  • < 0: no real roots

🎯 Exam Tip:
Sketch the graph to help visualize inequalities like x² – 4x – 5 > 0.

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🔣 6. Polynomials, Identities, and Theorems

When to Use:
When simplifying or factoring high-degree expressions.

How to Use:

• Apply Remainder Theorem:
  f(a) gives the remainder of dividing f(x) by (x – a)

• Factor Theorem:
  If f(a) = 0, then (x – a) is a factor.

🎯 Exam Tip:
When given a cubic polynomial, find one factor using Factor Theorem, then long divide.

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🧩 7. Partial Fractions

When to Use:
For integrating rational expressions.

How to Use: Break down:

• Distinct Linear Factors:
  A/(x + a) + B/(x + b)

• Repeated Factors:
  A/(x + a) + B/(x + a)²

• Irreducible Quadratic:
  (Ax + B)/(x² + b)

🎯 Exam Tip:
After splitting, plug in values of x that simplify equations to solve for A and B fast!

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📊 8. Modulus Function & Graphs

When to Use:
Equations or graphs with |x|.

How to Use:

• Understand that:
  |x| = x if x ≥ 0
  |x| = -x if x < 0

🎯 Exam Tip:
Split the modulus into cases when solving equations or sketching graphs.

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🔁 9. Binomial Expansion

When to Use:
For expressions like (a + b)^n

How to Use:

• Use combinations:
  nCr = n! / [r!(n–r)!]

• General Term:
  T_r+1 = nCr * a^(n–r) * b^r

🎯 Exam Tip:
Need the x⁵ term? Match the powers and solve for r!

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📐 10. Coordinate Geometry

When to Use:
When finding distances, gradients, midpoints.

How to Use:

• Midpoint: (x₁+x₂)/2, (y₁+y₂)/2

• Gradient: (y₂ - y₁)/(x₂ - x₁)

• Distance: √[(x₂ – x₁)² + (y₂ – y₁)²]

🎯 Exam Tip:
Use gradient relationships to show perpendicularity or parallel lines!

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📏 11. Geometry: Triangles, Circles & Angles

Key Concepts to Use:

• Triangle area with coordinates:
  Area = 1/2 |x₁(y₂–y₃) + x₂(y₃–y₁) + x₃(y₁–y₂)|

• Circle equation:
  (x – a)² + (y – b)² = r²

• Important Angle Properties:

  • ∠ in semicircle = 90°

  • ∠ at centre = 2 ∠ at circumference

🎯 Exam Tip:
Whenever you see circles or radii, mark all given lengths & angles immediately.

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📚 12. Trigonometry & Graphs

When to Use:
Whenever dealing with angles, triangles, or periodic functions.

How to Use:

• Special Angles: Memorize exact values for 30°, 45°, 60°

• Trig Identities to Memorize:

  • sin²θ + cos²θ = 1

  • 1 + tan²θ = sec²θ

Graph Basics:

• sin, cos = amplitude ±1

• tan = asymptotes at 90° intervals

🎯 Exam Tip:
To solve equations like 2sinθ = √3, find principal angle and check quadrant.

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🧠 13. Trig Equations, Identities, and R-Formulae

When to Use:
For expressions like a sin θ + b cos θ.

How to Use:

• Convert to single sine/cosine using:
  R = √(a² + b²), tan α = b/a

• Double Angle Formulae:

  • sin 2A = 2 sin A cos A

  • cos 2A = cos² A – sin² A

🎯 Exam Tip:
Always simplify expressions before solving — especially in MCQ!

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📈 14. Differentiation: Rules, Tangents & Rates

When to Use:
Whenever you need slope, turning points, or rates of change.

How to Use:

• Chain Rule: dy/dx = dy/du * du/dx

• Product Rule: (uv)' = u'v + uv'

• Quotient Rule: (u/v)' = (v u' – u v') / v²

🎯 Exam Tip:
Turning point = where dy/dx = 0. Use second derivative to check max/min!

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⏱️ 15. Integration & Area Under Curves

When to Use:
Reverse of differentiation. Used to find area under curves.

How to Use:

• Basic: ∫ xⁿ dx = xⁿ⁺¹ / (n+1)

• Definite Integrals:
  ∫[a to b] f(x) dx = F(b) – F(a)

• Area under x-axis? Take absolute value!

🎯 Exam Tip:
Watch the sign of area — under x-axis areas are negative unless absolute.

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🚗 16. Kinematics (Motion Math)

When to Use:
For velocity, acceleration, displacement problems.

How to Use:

• v = ds/dt, a = dv/dt

• Displacement = ∫ v dt

🎯 Exam Tip:
When a particle is at rest, v = 0. Use that to find when or where it stops.

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🔑 Final Revision Tips: How to Master the A-Math Formula Sheet

✅ Print it out — Stick it next to your study space.
✅ 10-Minute Rule — Revise one chapter a day for 10 minutes.
✅ Flashcard It — Turn formulas into flashcards.
✅ Use in Practice — Apply every formula while doing exam papers.
✅ Keep It Updated — Highlight formulas you forget or misuse during practices.

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📌 Get Your FREE A-Math Formula Sheet Now!

You can download Sophia Education’s full A-Math Summary Notes for free — covering 40+ subtopics in a single glance.

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✨ Why Students Love This Formula Sheet

💡 Super-condensed revision
⏱️ Saves time in last-minute prep
🎯 Focuses only on what you NEED to score
🧠 Perfect for concept recall & practice

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📞 Call or Whatsapp us today to enquire more!
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🎯 Conclusion

The A-Math Formula Sheet isn’t just a study tool — it’s your exam survival guide. Use it wisely, practice strategically, and you’ll walk into your O-Level confident and ready to crush it. 💪📘

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