Important Formulae for RRB Railway Exams

Master all essential mathematical and scientific formulae required for RRB railway examinations with our comprehensive collection organized by subject and topic for easy reference and quick revision.

Mathematics Formulae

📊 Arithmetic and Number System

Basic Operations and Properties

• Commutative Property:
  • Addition: a + b = b + a
  • Multiplication: a × b = b × a
• Associative Property:
  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a × b) × c = a × (b × c)
• Distributive Property: a × (b + c) = a × b + a × c

Percentage and Ratio

• Percentage Conversion:
  • Decimal to Percent: Multiply by 100
  • Fraction to Percent: (Numerator/Denominator) × 100
• Percentage Increase/Decrease:
  • Increase: Original × (1 + %increase/100)
  • Decrease: Original × (1 - %decrease/100)
• Ratio: a:b = a/b
• Proportion: a:b = c:d ⇒ a × d = b × c

Profit and Loss

• Profit: SP - CP (where SP > CP)
• Loss: CP - SP (where CP > SP)
• Profit Percentage: (Profit/CP) × 100
• Loss Percentage: (Loss/CP) × 100
• Discount: MP - SP
• Discount Percentage: (Discount/MP) × 100

Simple and Compound Interest

• Simple Interest: SI = (P × R × T)/100
• Amount: A = P + SI
• Compound Interest: A = P(1 + R/100)^T
• CI Amount: CI = A - P

Time and Work

• Work: Work = Rate × Time
• Rate: Rate = Work/Time
• Time: Time = Work/Rate
• Work Efficiency: If A can do work in x days and B in y days:
  • Together: (1/x + 1/y) work per day
  • Time together: xy/(x + y) days

Time and Distance

• Speed: Speed = Distance/Time
• Distance: Distance = Speed × Time
• Time: Time = Distance/Speed
• Average Speed: Total Distance/Total Time
• Relative Speed:
  • Same direction: |v₁ - v₂|
  • Opposite direction: v₁ + v₂

🔢 Algebra

Quadratic Equations

• Standard Form: ax² + bx + c = 0
• Discriminant: Δ = b² - 4ac
• Roots Formula: x = [-b ± √(b² - 4ac)]/2a
• Sum of Roots: α + β = -b/a
• Product of Roots: α × β = c/a

Progressions

• Arithmetic Progression (AP):
  • nth term: aₙ = a + (n-1)d
  • Sum of n terms: Sₙ = n/2[2a + (n-1)d]
  • Sum of first n natural numbers: Sₙ = n(n+1)/2
• Geometric Progression (GP):
  • nth term: aₙ = ar^(n-1)
  • Sum of n terms: Sₙ = a(rⁿ - 1)/(r - 1) for r ≠ 1
  • Sum to infinity: S∞ = a/(1 - r) for |r| < 1

📐 Geometry and Mensuration

Plane Geometry

• Triangle:
  • Area: (1/2) × base × height
  • Perimeter: Sum of all sides
  • Heron’s Formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2
• Rectangle:
  • Area: length × breadth
  • Perimeter: 2(length + breadth)
  • Diagonal: √(length² + breadth²)
• Square:
  • Area: side²
  • Perimeter: 4 × side
  • Diagonal: side × √2
• Circle:
  • Area: πr²
  • Circumference: 2πr
  • Diameter: 2r
• Parallelogram:
  • Area: base × height
  • Perimeter: 2(side₁ + side₂)

Solid Geometry

• Cube:
  • Volume: side³
  • Surface Area: 6 × side²
  • Diagonal: side × √3
• Cuboid:
  • Volume: length × breadth × height
  • Surface Area: 2(lb + bh + hl)
  • Diagonal: √(l² + b² + h²)
• Sphere:
  • Volume: (4/3)πr³
  • Surface Area: 4πr²
• Cylinder:
  • Volume: πr²h
  • Curved Surface Area: 2πrh
  • Total Surface Area: 2πr(h + r)
• Cone:
  • Volume: (1/3)πr²h
  • Curved Surface Area: πrl (where l = √(r² + h²))
  • Total Surface Area: πr(l + r)

📈 Trigonometry

Basic Trigonometric Ratios

• sin θ = Opposite/Hypotenuse
• cos θ = Adjacent/Hypotenuse
• tan θ = Opposite/Adjacent = sin θ/cos θ
• cot θ = Adjacent/Opposite = cos θ/sin θ
• sec θ = Hypotenuse/Adjacent = 1/cos θ
• cosec θ = Hypotenuse/Opposite = 1/sin θ

Important Trigonometric Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• sin(90° - θ) = cos θ
• cos(90° - θ) = sin θ
• tan(90° - θ) = cot θ

Trigonometric Values for Common Angles

Physics Formulae

⚛️ Mechanics

Motion and Kinematics

• Average Velocity: v_avg = Δx/Δt
• Average Acceleration: a_avg = Δv/Δt
• Equations of Motion:
  • v = u + at
  • s = ut + (1/2)at²
  • v² = u² + 2as
  • s = vt - (1/2)at²
• Distance: d = speed × time

Force and Newton’s Laws

• Newton’s Second Law: F = ma
• Weight: W = mg (where g = 9.8 m/s²)
• Momentum: p = mv
• Impulse: J = F × Δt = Δp
• Friction: f = μN (where μ = coefficient of friction)

Work, Energy, and Power

• Work: W = F × d × cos θ
• Kinetic Energy: KE = (1/2)mv²
• Potential Energy: PE = mgh
• Total Mechanical Energy: E = KE + PE
• Power: P = W/t = F × v
• Efficiency: η = (Output/Input) × 100%

Circular Motion

• Angular Velocity: ω = v/r = 2π/T
• Centripetal Acceleration: a_c = v²/r = ω²r
• Centripetal Force: F_c = mv²/r = mω²r
• Period: T = 2π/ω
• Frequency: f = 1/T = ω/2π

🌊 Waves and Sound

Wave Properties

• Wave Speed: v = fλ (frequency × wavelength)
• Time Period: T = 1/f
• Angular Frequency: ω = 2πf
• Wave Number: k = 2π/λ
• Wave Equation: y = A sin(kx - ωt)

Sound

• Speed of Sound: v = 331 + 0.6T m/s (T in °C)
• Doppler Effect:
  • Source moving: f’ = f[v/(v ± v_s)]
  • Observer moving: f’ = f[(v ± v_o)/v]
• Intensity Level: β = 10 log₁₀(I/I₀) dB

🔥 Thermodynamics

Heat and Temperature

• Heat Capacity: Q = mcΔT
• Latent Heat: Q = mL
• Thermal Expansion:
  • Linear: ΔL = L₀αΔT
  • Volume: ΔV = V₀βΔT
• Gas Laws:
  • Boyle’s Law: P₁V₁ = P₂V₂ (constant T)
  • Charles’s Law: V₁/T₁ = V₂/T₂ (constant P)
  • Gay-Lussac’s Law: P₁/T₁ = P₂/T₂ (constant V)
  • Ideal Gas Law: PV = nRT

💡 Electricity and Magnetism

Electrostatics

• Coulomb’s Law: F = k(q₁q₂)/r²
• Electric Field: E = F/q = k(Q)/r²
• Electric Potential: V = k(Q)/r
• Capacitance: C = Q/V
• Energy Stored: U = (1/2)CV²

Current Electricity

• Ohm’s Law: V = IR
• Resistance: R = ρ(L/A)
• Power: P = VI = I²R = V²/R
• Energy: E = Pt = VIt
• Kirchhoff’s Laws:
  • Current Law: ΣI_in = ΣI_out
  • Voltage Law: ΣV = 0 around closed loop

Magnetism

• Magnetic Force: F = qvB sin θ
• Force on Current-carrying Conductor: F = BIL sin θ
• Magnetic Flux: Φ = BA cos θ
• Faraday’s Law: ε = -N(ΔΦ/Δt)

Chemistry Formulae

⚗️ Stoichiometry

Mole Concept

• Number of Moles: n = mass/molar mass
• Number of Particles: N = n × N_A (where N_A = 6.022 × 10²³)
• Molar Mass: M = mass/number of moles
• Molar Volume: V_m = 22.4 L at STP

Concentration

• Molarity: M = moles of solute/volume of solution (L)
• Molality: m = moles of solute/mass of solvent (kg)
• Normality: N = equivalents of solute/volume of solution (L)
• Mass Percentage: % = (mass of solute/mass of solution) × 100

🧪 Chemical Reactions

Rate of Reaction

• Rate Law: Rate = k[A]^m[B]^n
• Order of Reaction: Overall order = m + n
• Half-Life: t₁/₂ = 0.693/k (for first-order reactions)

Equilibrium

• Equilibrium Constant: K = [Products]^coefficients/[Reactants]^coefficients
• Acid Dissociation: Ka = [H⁺][A⁻]/[HA]
• Base Dissociation: Kb = [BH⁺][OH⁻]/[B]
• pH: pH = -log[H⁺]
• pOH: pOH = -log[OH⁻]
• pH + pOH = 14 (at 25°C)

Engineering Formulae

⚙️ Mechanical Engineering

Strength of Materials

• Stress: σ = Force/Area
• Strain: ε = ΔL/L₀
• Young’s Modulus: E = σ/ε
• Shear Stress: τ = Shear Force/Area
• Shear Strain: γ = tan φ
• Modulus of Rigidity: G = τ/γ
• Bulk Modulus: K = -P/(ΔV/V₀)

Fluid Mechanics

• Pressure: P = F/A
• Hydrostatic Pressure: P = ρgh
• Buoyant Force: F_b = ρVg
• Continuity Equation: A₁v₁ = A₂v₂
• Bernoulli’s Equation: P + (1/2)ρv² + ρgh = constant
• Reynolds Number: Re = ρvD/μ

Thermodynamics

• First Law: ΔU = Q - W
• Work Done: W = PΔV
• Efficiency of Heat Engine: η = 1 - T_cold/T_hot
• Carnot Efficiency: η_carnot = 1 - T_cold/T_hot

⚡ Electrical Engineering

Electrical Circuits

• Resistance in Series: R_total = R₁ + R₂ + … + Rₙ
• Resistance in Parallel: 1/R_total = 1/R₁ + 1/R₂ + … + 1/Rₙ
• Capacitance in Series: 1/C_total = 1/C₁ + 1/C₂ + … + 1/Cₙ
• Capacitance in Parallel: C_total = C₁ + C₂ + … + Cₙ
• Inductance in Series: L_total = L₁ + L₂ + … + Lₙ
• Inductance in Parallel: 1/L_total = 1/L₁ + 1/L₂ + … + 1/Lₙ

AC Circuits

• Impedance (RLC Series): Z = √[R² + (X_L - X_C)²]
• Inductive Reactance: X_L = 2πfL
• Capacitive Reactance: X_C = 1/(2πfC)
• Resonant Frequency: f₀ = 1/(2π√LC)
• Power Factor: cos φ = R/Z

Transformers

• Voltage Ratio: V₁/V₂ = N₁/N₂
• Current Ratio: I₁/I₂ = N₂/N₁
• Efficiency: η = (Output Power/Input Power) × 100%

Conversion Tables

📏 Length Conversions

• 1 inch = 2.54 cm
• 1 foot = 30.48 cm = 12 inches
• 1 yard = 91.44 cm = 3 feet
• 1 mile = 1.609 km = 5280 feet
• 1 km = 1000 m = 0.621 miles

⚖️ Mass/Weight Conversions

• 1 kg = 1000 g = 2.205 pounds
• 1 pound = 453.6 g = 0.4536 kg
• 1 tonne = 1000 kg = 2204.6 pounds

🌡️ Temperature Conversions

• Celsius to Fahrenheit: F = (9/5)C + 32
• Fahrenheit to Celsius: C = (5/9)(F - 32)
• Celsius to Kelvin: K = C + 273.15
• Kelvin to Celsius: C = K - 273.15

💧 Volume Conversions

• 1 liter = 1000 ml = 1000 cm³
• 1 gallon = 3.785 liters
• 1 cubic meter = 1000 liters

How to Use These Formulae

1. Regular Revision: Review formulae daily
2. Practice Application: Solve problems using these formulae
3. Memory Techniques: Use mnemonics and patterns
4. Group Study: Discuss with peers for better retention
5. Mock Tests: Apply formulae in test conditions
6. Formula Sheets: Create personal quick-reference sheets

These formulae cover all essential topics for RRB railway examinations. Regular practice and application of these formulae will help you solve problems quickly and accurately during the exam.