Formula Sheet

---
target: 495+/500
author: Senior Sibling
name: formula-sheet
description: >-
  ALL 12th Grade Formulas (PCMB/PCMC). 
  The single most important revision resource for numerical-heavy units.
---

# Formula Master Sheet — CBSE Grade 12

## 📐 Mathematics Formulas

### Relations, Functions & Algebra
```latex
Matrix Inverse: A^{-1} = (1/|A|) * adj(A)
Adjoint Property: A * adj(A) = |A| I
Area of Triangle (Determinants): Δ = 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
```

### Calculus (Ch 5-9)
```latex
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Logarithmic: d/dx[u^v] = u^v [v/u * du/dx + log(u) * dv/dx]
Integration by Parts: ∫ uv dx = u∫v dx - ∫[u'∫v dx] dx (ILATE rule)
King Property: ∫_0^a f(x) dx = ∫_0^a f(a-x) dx
Linear Diff Eq: dy/dx + Py = Q → I.F. = e^{∫P dx} → y * I.F. = ∫(Q * I.F.) dx + C
```

### Vectors & 3D Geometry
```latex
Dot Product: a.b = |a||b| cosθ
Cross Product: a x b = |a||b| sinθ n̂
Shortest Distance: d = |(b1 x b2).(a2 - a1)| / |b1 x b2|
Angle between lines: cosθ = |b1.b2| / (|b1||b2|)
```

### Probability
```latex
Conditional: P(A|B) = P(A ∩ B) / P(B)
Multiplication Theorem: P(A ∩ B) = P(A) * P(B|A)
Bayes' Theorem: P(Ei|A) = [P(Ei)P(A|Ei)] / Σ[P(Ej)P(A|Ej)]
```

---

## 🔭 Physics Formulas

### Electrostatics & Current
```latex
Coulomb's Law: F = (1/4πε₀) * q1q2/r²
Electric Field (Axial): E = (1/4πε₀) * 2p/r³
Electric Field (Equatorial): E = (1/4πε₀) * p/r³
Gauss Law: Φ = q/ε₀
Potential: V = (1/4πε₀) * q/r
Capacitance: C = ε₀A/d (With dielectric K: C = Kε₀A/d)
Current Density: J = I/A = n e v_d
Drift Velocity: v_d = eEτ/m
Resistance: R = ρl/A = (m/ne²τ) * l/A
Potentiometer: E1/E2 = l1/l2
```

### Magnetism & EMI/AC
```latex
Biot-Savart: dB = (μ₀/4π) * I dl sinθ / r²
Magnetic Field (Circular loop center): B = μ₀I / 2R
Ampere's Law: ∫ B.dl = μ₀I
Lorentz Force: F = q(E + v x B)
Magnetic Flux: Φ = B A cosθ
Faraday's Law: ε = -dΦ/dt
Self Induction: ε = -L dI/dt
Impedance (LCR): Z = √[R² + (X_L - X_C)²]
Resonant Freq: f = 1 / [2π√(LC)]
Transformer: V_s/V_p = N_s/N_p = I_p/I_s
```

### Optics & Modern Physics
```latex
Lens Maker: 1/f = (n2/n1 - 1) [1/R1 - 1/R2]
Magnification (Microscope): M = (L/f₀) * (D/f_e)
Fringe Width (YDSE): β = λD/d
Photoelectric Eq: K_max = hν - Φ₀
de Broglie: λ = h/mv = h/√(2mE)
Bohr Radius: rₙ = (ε₀h²n²) / (πmZe²)
Energy (n-th state): Eₙ = -13.6 Z²/n² eV
```

---

## 🧪 Chemistry Formulas

### Physical Chemistry
```latex
Raoult's Law: P_A = P⁰_A * χ_A
Elevation in BP: ΔT_b = i * K_b * m
Nernst Eq: E_cell = E⁰_cell - (0.0591/n) log Q (at 298K)
Molar Conductivity: Λ_m = (κ * 1000) / M
Rate Constant (1st Order): k = (2.303/t) log([R]₀/[R])
Half-life (1st Order): t_{1/2} = 0.693/k
Arrhenius Eq: log(k2/k1) = (Ea / 2.303R) * [(T2-T1)/(T1T2)]
```