MATHS – SHORT NOTES

1. Algebra – Identities

• (a + b)² = a² + b² + 2ab

• (a – b)² = a² + b² – 2ab

• a² – b² = (a + b)(a – b)

• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

2. Trigonometry

Basic Ratios

• sinθ = Perpendicular/Hypotenuse

• cosθ = Base/Hypotenuse

• tanθ = sinθ / cosθ

Identities

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

Important Values

• sin 0° = 0

• sin 30° = 1/2

• sin 45° = √2/2

• sin 60° = √3/2

• sin 90° = 1

3. Coordinate Geometry

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

• Midpoint formula:
  ((x₁ + x₂)/2 , (y₁ + y₂)/2)

• Slope: (y₂ − y₁) / (x₂ − x₁)

• Equation of a line:
  y = mx + c

4. Calculus (Basics)

Derivatives

• d/dx (xⁿ) = nxⁿ⁻¹

• d/dx (sin x) = cos x

• d/dx (cos x) = −sin x

• d/dx (eˣ) = eˣ

Integration

• ∫ xⁿ dx = xⁿ⁺¹/(n + 1)

• ∫ 1 dx = x

• ∫ eˣ dx = eˣ + C

5. Sets & Relations

• A ⊂ B → A is subset of B

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

6. Probability

• Probability = (Favorable outcomes) / (Total outcomes)

• 0 ≤ P ≤ 1

• P(A’) = 1 − P(A)

7. Vectors

• Magnitude of vector a = √(a₁² + a₂² + a₃²)

• Dot product: a·b = |a||b|cosθ

• Cross product: |a × b| = |a||b|sinθ

8. Matrices

• Matrix addition: Add element-wise

• Matrix multiplication: Row × Column

• Identity matrix (I): diagonal = 1

9. Determinants

For 2×2 matrix:
|a b|
|c d| = ad − bc

10. Statistics

• Mean = Σx / n

• Median → Middle value

• Mode → Most repeated value