Formula Sheet

---
name: formula-sheet
description: ALL formulas across Math and Science in one file. The single most important revision resource — students who memorize these formulas can attempt ANY numerical.
---

# Formula Quick Reference — Board Exam Lifeline

## Mathematics Formulas

### Real Numbers
```
HCF × LCM = Product of two numbers
Euclid's Division Lemma: a = bq + r, 0 ≤ r < b
```

### Polynomials
```
Linear: ax + b, zero = -b/a
Quadratic: ax² + bx + c
  Sum of zeroes (α + β) = -b/a
  Product of zeroes (αβ) = c/a
Cubic: ax³ + bx² + cx + d
  α + β + γ = -b/a
  αβ + βγ + γα = c/a
  αβγ = -d/a
```

### Quadratic Equations
```
Quadratic formula: x = (-b ± √(b²-4ac)) / 2a
Discriminant: D = b² - 4ac
  D > 0 → two distinct real roots
  D = 0 → two equal real roots
  D < 0 → no real roots
```

### Arithmetic Progression
```
nth term: aₙ = a + (n-1)d
Sum of n terms: Sₙ = n/2 [2a + (n-1)d] = n/2 [a + l]
  where a = first term, d = common difference, l = last term
```

### Coordinate Geometry
```
Distance: √[(x₂-x₁)² + (y₂-y₁)²]
Section formula: ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂))
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Area of triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
```

### Trigonometry
```
sin θ = opposite/hypotenuse    cosec θ = 1/sin θ
cos θ = adjacent/hypotenuse    sec θ = 1/cos θ
tan θ = opposite/adjacent      cot θ = 1/tan θ

Standard values:
       0°    30°    45°    60°    90°
sin    0    1/2    1/√2   √3/2    1
cos    1    √3/2   1/√2   1/2     0
tan    0    1/√3    1     √3      ∞

Identities:
  sin²θ + cos²θ = 1
  1 + tan²θ = sec²θ
  1 + cot²θ = cosec²θ
```

### Mensuration
```
Circle: Area = πr², Circumference = 2πr
Sector: Area = (θ/360)πr², Arc length = (θ/360)×2πr

Cylinder: CSA = 2πrh, TSA = 2πr(h+r), Volume = πr²h
Cone: CSA = πrl, TSA = πr(l+r), Volume = ⅓πr²h, l = √(r²+h²)
Sphere: CSA = TSA = 4πr², Volume = ⁴⁄₃πr³
Hemisphere: CSA = 2πr², TSA = 3πr², Volume = ⅔πr³
Frustum: Volume = ⅓πh(r₁² + r₂² + r₁r₂)
```

### Statistics
```
Mean (direct): x̄ = Σfᵢxᵢ / Σfᵢ
Mean (assumed): x̄ = a + Σfᵢdᵢ/Σfᵢ (where dᵢ = xᵢ - a)
Mean (step-deviation): x̄ = a + h(Σfᵢuᵢ/Σfᵢ) (where uᵢ = (xᵢ-a)/h)
Median = l + [(n/2 - cf)/f] × h
Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h
```

### Probability
```
P(E) = Number of favourable outcomes / Total outcomes
P(E) + P(Ē) = 1
0 ≤ P(E) ≤ 1
```

## Physics Formulas

### Electricity
```
V = IR (Ohm's law)
R = ρl/A (resistivity)
Series: R = R₁ + R₂ + R₃
Parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃
Power: P = VI = I²R = V²/R
Energy: E = Pt = VIt
1 kWh = 3.6 × 10⁶ J
```

### Light
```
Mirror formula: 1/v + 1/u = 1/f
Magnification: m = -v/u = hᵢ/hₒ
Lens formula: 1/v - 1/u = 1/f
Power of lens: P = 1/f (dioptre, f in metres)
Snell's law: n = sin i / sin r
```

## Chemistry Formulas
```
Balanced equations must include state symbols: (s), (l), (g), (aq)
pH scale: 0-14 (7 = neutral, <7 = acidic, >7 = basic)
```