📐 Leaving Cert Maths

Master Maths 💪

Complete HL & OL coverage with clear notes, worked examples, and exam-focused practice questions.

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Higher Level Topics

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Algebra

  • • Equations & Inequalities
  • • Sequences & Series
  • • Complex Numbers
  • • Induction
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Calculus

  • • Differentiation
  • • Integration
  • • Rates of Change
  • • Area & Volume
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Trigonometry

  • • Trig Functions
  • • Identities
  • • Equations
  • • Applications
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Geometry

  • • Coordinate Geometry
  • • Synthetic Geometry
  • • Transformations
  • • Vectors
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Probability & Stats

  • • Probability Rules
  • • Distributions
  • • Statistics & Inference
  • • Data Analysis
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Number Theory

  • • Number Systems
  • • Modular Arithmetic
  • • Diophantine Equations
  • • Cryptography Basics
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Exam Papers & Solutions

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Past Papers

2010–2024 papers for HL & OL

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Algebra — Core Concepts

Key algebra skills: factorising quadratics, solving equations/inequalities, sequences & series, and complex numbers.

Example 1 — Solve 2x² − 7x + 3 = 0

Factorise: (2x − 1)(x − 3) = 0 → x = ½ or x = 3.

Example 2 — Find the nth term of 3, 7, 11, …

This is an arithmetic sequence with a₁ = 3, d = 4 → aₙ = 3 + (n − 1)×4 = 4n − 1.

Example 3 — Solve |2x − 3| = 5

2x − 3 = 5 or 2x − 3 = −5 → x = 4 or x = −1.

Example 4 — Find the sum of first 10 terms of aₙ = 2n + 1

Sₙ = n/2 × (first + last) = 10/2 × (3 + 21) = 5×24 = 120.

Example 5 — Simplify (1 + i)(1 − i)

(1 + i)(1 − i) = 1 − i² = 2.

Calculus — Differentiation & Integration

Understand slopes, rates of change, and areas under curves.

Example 1 — Differentiate f(x) = x³ − 6x² + 9x

f′(x) = 3x² − 12x + 9.

Example 2 — Find ∫(3x² − 2x + 1) dx

= x³ − x² + x + C.

Example 3 — Find stationary points of f(x)=x³−6x²+9x

f′(x)=3x²−12x+9=0 → x=1,3 → local max/min points.

Example 4 — Find derivative of y = sin(x) + cos(x)

y′ = cos(x) − sin(x).

Example 5 — Area under y=x² from 0 to 2

∫₀² x² dx = [x³/3]₀² = 8/3.

Trigonometry — Identities & Equations

Learn how to simplify and solve trigonometric expressions.

Example 1 — Solve sin(θ)=½ for 0°≤θ<360°

θ=30°,150°.

Example 2 — Prove sin²θ + cos²θ = 1

From unit circle definition of sine and cosine → identity holds for all θ.

Example 3 — Solve tan(θ)=√3

θ=60°, 240°.

Example 4 — Find amplitude of y=3sin(x)

Amplitude = 3.

Example 5 — Simplify sin(2θ)/cos(θ)

= 2sin(θ).

Geometry — Coordinate & Vectors

Covering coordinate geometry, transformations, and vector work.

Example 1 — Find distance between A(1,2) and B(5,5)

d=√((5−1)²+(5−2)²)=5.

Example 2 — Find midpoint of (2,3) & (6,7)

M=(4,5).

Example 3 — Equation of line through (1,2) slope 3

y−2=3(x−1) → y=3x−1.

Example 4 — Equation of circle center (0,0) radius 5

x²+y²=25.

Example 5 — Vector magnitude of (3,4)

|v|=√(3²+4²)=5.

Probability & Statistics — Essentials

Covers probability laws, distributions, and statistics.

Example 1 — Toss a coin: P(Head)

1/2.

Example 2 — Rolling a die: P(even)

3/6 = ½.

Example 3 — E(X) for X={1,2,3} with p={0.2,0.5,0.3}

E(X)=2.1.

Example 4 — Variance formula

Var(X)=E(X²)−[E(X)]².

Example 5 — Normal distribution: z=(x−μ)/σ

Converts any score to a standard normal variable.

Number Theory — Modular Arithmetic

Covers modular arithmetic, inverses, and divisibility.

Example 1 — 17 ≡ ? mod 5

17 ≡ 2 mod 5.

Example 2 — Solve 3x ≡ 1 (mod 7)

x ≡ 5 (mod 7).

Example 3 — Check if 91 is prime

91 = 7×13 → not prime.

Example 4 — Find gcd(60,24)

12.

Example 5 — Find inverse of 5 mod 12

5×5=25≡1 (mod12) → inverse=5.