JEE Main 2026 Complete Formula Sheet: Physics, Chemistry, Maths
Revision Resource•Updated March 2026•VRSAM Team
Success in JEE Main often comes down to two things: conceptual clarity and the speed at which you can recall standard results. Use these one-tap copy buttons to save formulas to your notes.
Physics Formulas
Mechanics
Kinematics
| Velocity | v = u + at | |
| Displacement | s = ut + ½at² | |
| Velocity-Displacement | v² = u² + 2as | |
| Nth Second | Sₙ = u + a/2(2n - 1) | |
| Time of Flight | T = (2u sinθ)/g | |
| Max Height | H = (u² sin²θ)/2g | |
| Range | R = (u² sin2θ)/g | |
| Trajectory | y = x tanθ - (gx²)/(2u²cos²θ) |
Work, Energy & Power
| Work | W = F⃗ · d⃗ = Fs cosθ | |
| Kinetic Energy | KE = ½mv² | |
| Potential Energy | PE = mgh | |
| Power | P = W/t = F⃗ · v⃗ | |
| Work-Energy Theorem | W_net = ΔK | |
| Force from PE | F = -dU/dr | |
| Coeff. of Restitution | e = (v₂ - v₁)/(u₁ - u₂) | |
| Elastic Collision | e = 1 |
Gravitation
| Force | F = G(m₁m₂)/r² | |
| g at height h | g' = g(1 - 2h/R) | |
| Escape Velocity | vₑ = √(2GM/R) | |
| Orbital Velocity | v₀ = √(GM/R) |
Uniform Circular Motion
| Angular Displacement | Δθ = ωΔt | |
| Linear Velocity | v = Rω | |
| Centripetal Accel. | aᶜ = v²/R = ω²R = 4π²ν²R | |
| Angular Velocity | ω = dθ/dt | |
| Angular Accel. | α = dω/dt = ω(dω/dθ) | |
| Velocity Vector | v⃗ = ω⃗ × r⃗ | |
| Tangential Accel. | aₜ = dV/dt = r(dω/dt) | |
| Radial Accel. | aᵣ = V²/r = ω²r | |
| Concave Normal | N = mg cosθ + mv²/r | |
| Convex Normal | N = mg cosθ - mv²/r | |
| Safe Speed | V_safe ≤ √(μgr) | |
| Max Angular Speed | ω_max = √(μg/r) | |
| Banking Angle | tanθ = v²/(rg) | |
| V_max (banked) | V_max = [rg(μ + tanθ)/(1 - μtanθ)]^(1/2) | |
| V_min (banked) | V_min = [rg(tanθ - μ)/(1 + μtanθ)]^(1/2) |
Centre of Mass
| CM Position | r⃗_cm = Σmᵢr⃗ᵢ / Σmᵢ | |
| Rectangular Plate | rₓ = B/2, rᵧ = L/2 | |
| Triangular Plate | rᶜ = h/3 | |
| Semi-circular Ring | rᵧ = 2R/π | |
| Semi-circular Disc | rᵧ = 4R/(3π) | |
| Hemispherical Shell | rᵧ = R/2 | |
| Solid Hemisphere | rᵧ = 3R/8 | |
| Circular Cone | rᵧ = h/4 | |
| Hollow Cone | rᵧ = h/3 |
Linear Momentum
| Momentum | p⃗ = mv⃗ | |
| Conservation | m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | |
| Newton's 2nd Law | F_net = dp⃗/dt = ma | |
| KE-Momentum | p² = 2mK | |
| Perfectly Inelastic | m₁v₁ᵢ + m₂v₂ᵢ = (m₁+m₂)vf |
Friction
| Kinetic Friction | Fₖ = μₖR | |
| Static Friction | Fₛ = μₛR | |
| Normal (level) | R = mg | |
| Normal (incline) | R = mg cosθ |
Fluid Mechanics
| Pressure | P = F/A | |
| Hydraulic Force | F = (A/a)f | |
| Tilted Surface | tanθ = a₀/g | |
| Continuity Eq. | a₁v₁ = a₂v₂ | |
| Bernoulli's Eq. | P/(ρg) + v²/(2g) + Z = const | |
| Efflux Velocity | v = √(2gh / (1 - (A₂/A₁)²)) | |
| Stokes Force | F = 6πηrv | |
| Terminal Velocity | vₜ = 2r²(ρ-σ)g/(9η) |
Hooke's Law & Elasticity
| Hooke's Law | F = -kx | |
| Stress | Stress = F/A | |
| Strain | Strain = ΔL/L | |
| Young's Modulus | Y = (F/A)/(ΔL/L) = FL/(AΔL) | |
| Shear Modulus | G = (F/A)/(Δx/h) |
Electromagnetism
Electrostatics
| Coulomb's Law | F⃗ = (1/4πε₀)(q₁q₂/r²)r̂ | |
| Electric Field | E⃗ = F⃗/q₀ | |
| Ring (axis) | E = KQx/(R²+x²)^(3/2) | |
| Disc | E = (σ/2ε₀)[1 - x/√(R²+x²)] | |
| Potential | V = (1/4πε₀)(q/r) | |
| Work Done | W = q(Vₐ - V_B) | |
| PE of System | U = (1/4πε₀)(q₁q₂/r) | |
| Gauss's Law | ∮ E⃗·dA⃗ = q/ε₀ | |
| Dipole Moment | p⃗ = qd⃗ | |
| Dipole Potential | V = p cosθ/(4πε₀r²) | |
| Torque | τ⃗ = p⃗ × E⃗ |
Capacitance
| Capacitance | C = Q/V | |
| Parallel Plate | C = ε₀A/d | |
| Spherical | C = 4πε₀(rₐr_b)/(r_b - rₐ) | |
| Cylindrical (per length) | C = 2πε₀/ln(b/a) | |
| Electric Field | E = σ/ε₀ = V/d | |
| Energy Stored | U = ½CV² = Q²/(2C) = QV/2 | |
| Energy Density | u = ½ε₀εᵣE² | |
| Series | 1/C_eq = 1/C₁ + 1/C₂ + ... + 1/Cₙ | |
| Parallel | C_eq = C₁ + C₂ + ... + Cₙ | |
| Charging | q = q₀(1 - e^(-t/τ)) | |
| Discharging | q = q₀ e^(-t/τ) |
Current Electricity
| Current | I = Δq/Δt = nAev_d | |
| Ohm's Law | V = IR | |
| Resistance | R = ρl/A | |
| Temp. Dependence | R = R₀(1 + αΔT) | |
| Power | P = VI = I²R = V²/R | |
| Heat | H = VIt = I²Rt | |
| Series | R_eq = R₁ + R₂ + ... | |
| Parallel | 1/R_eq = 1/R₁ + 1/R₂ + ... | |
| EMF (parallel cells) | E_eq = (ε₁/r₁ + ε₂/r₂ + ...)/(1/r₁ + 1/r₂ + ...) | |
| Ammeter Shunt | S = IgRg/I | |
| Voltmeter Resistance | Rs = V/Ig - Rg | |
| Kirchhoff's Laws | ΣI = 0 (junction); ΣIR = 0 (loop) | |
| Metre Bridge | x/R = l₁/(100 - l₁) | |
| Potentiometer | E₁/E₂ = l₁/l₂ | |
| Conductance | σ = 1/ρ; G = 1/R |
Magnetic Effect of Current
| Moving Charge Field | B⃗ = (μ₀/4π) q(v⃗×r⃗)/r³ | |
| Finite Wire | B = (μ₀I/4πr)(sinθ₁ + sinθ₂) | |
| Infinite Wire | B = μ₀I/(2πr) | |
| Loop (axis) | B = μ₀NIR²/[2(R²+x²)^(3/2)] | |
| Loop (centre) | B = μ₀NI/(2r) | |
| Solenoid | B = (μ₀NI/2)(cosθ₁ - cosθ₂) | |
| Force on Charge | F⃗ = q(v⃗ × B⃗) | |
| Force on Wire | F⃗ = I(l⃗ × B⃗) | |
| Magnetic Moment | M = NIA | |
| Torque | τ⃗ = M⃗ × B⃗ | |
| Axial Field | B = μ₀(2M)/(4πr³) | |
| Equatorial Field | B = μ₀M/(4πr³) | |
| General Point | Bp = (μ₀M/4πr³)√(1 + 3cos²θ) |
Ampere's Circuital Law
| Ampere's Law | ∮ B⃗·dl⃗ = μ₀I | |
| μ₀ value | μ₀ = 4π × 10⁻⁷ N/A² | |
| Wire Field | B = μ₀I/(2πr) | |
| Solenoid | BL = μ₀NI | |
| Thick Wire (inside) | B = μ₀Ir/(2πR²) | |
| Toroid | B = μ₀NI/(2πr) | |
| Force Between Wires | F/L = μ₀IₐI_B/(2πr) |
Electromagnetic Induction
| Flux | φ = ∫B⃗·dA⃗ | |
| Faraday's Law | E = -dφ/dt = -N(dφ/dt) | |
| Induced Current | I = E/R = (N/R)(dφ/dt) | |
| Self Induction | φ = LI; E = -L(dI/dt) | |
| Mutual Induction | e₂ = M(dI₁/dt); M = μ₀N₁N₂A/l |
Alternating Current
| RMS Value | f_rms = √(∫f(t)²dt / (t₂-t₁)) | |
| Avg Power | ⟨P⟩ = V_rms · I_rms · cosφ | |
| Impedance | Z = Vm/Im = V_rms/I_rms | |
| Inductive Reactance | X_L = ωL | |
| Capacitive Reactance | X_C = 1/(ωC) | |
| Resistive | I = Vm sinωt/R; ⟨P⟩ = V²_rms/R | |
| Capacitive | I = (Vm/X_C)cosωt; φ = 90°; ⟨P⟩ = 0 |
Electromagnetic Waves
| Gauss (Electric) | ∮ E⃗·dA⃗ = Q/ε₀ | |
| Gauss (Magnetic) | ∮ B⃗·dA⃗ = 0 | |
| Faraday | ∮ E⃗·dl⃗ = -dΦ_B/dt | |
| Ampere-Maxwell | ∮ B⃗·dl⃗ = μ₀i + μ₀ε₀(dΦ_E/dt) | |
| Speed of Light | c = 1/√(μ₀ε₀); E₀/B₀ = c |
Inductance
| Inductance | L = μN²A/l | |
| EMF | V = L(di/dt) | |
| Inductive Reactance | X_L = 2πfL |
Optics & Waves
Geometrical Optics
| Snell's Law | sini/sinr = n | |
| Refractive Index | n = c/v | |
| Lateral Shift | t·sin(i-r)/cosr | |
| Normal Shift | t(1 - 1/n) | |
| Critical Angle | n = 1/sinC | |
| Prism Formula | n = sin((A+δ)/2)/sin(A/2) | |
| Lens Maker's | 1/f = (n-1)(1/R₁ - 1/R₂) | |
| Power | P = 1/f | |
| Combination | 1/f = 1/f₁ + 1/f₂ |
Wave Physics
| Wave Equation | ∂²y/∂t² = v²(∂²y/∂x²) | |
| Wave Number | k = 2π/λ = ω/v | |
| Phase Diff. | Δφ = (2π/λ)Δx | |
| Speed | v = √(T/μ) | |
| Power | P = 2π²f²A²μv | |
| Intensity | I = 2π²f²A²ρv | |
| Speed of Sound | C = √(E/ρ) | |
| Loudness (dB) | L = 10 log₁₀(I/I₀) | |
| Intensity | I = P/(4πr²) | |
| Closed Pipe | f = (2n+1)v/(4l) | |
| Open Pipe | f = nv/(2l) | |
| Beats | Δf = f₁ - f₂ | |
| Doppler Effect | f' = f(v - v₀)/(v - vₛ) |
Wave Optics
| Path Diff. | Δd = d₂ - d₁ | |
| Constructive | Δd = kλ | |
| Destructive | Δd = (2k+1)λ/2 | |
| Film (constructive) | 2nt cosr = (n + ½)λ | |
| Film (destructive) | 2nt cosr = nλ | |
| Newton's Rings | r = √(kRλ) | |
| Grating (max) | d sinθ = kλ |
Modern Physics & Thermal
De Broglie & Atomic Physics
| De Broglie | λ = h/(mv) = h/√(2mKE) | |
| Radius | rₙ = (n²/Z)a₀; a₀ = 0.529 × 10⁻¹⁰ m | |
| Velocity | vₙ = (Z/n)v₀; v₀ = 2.19 × 10⁶ m/s | |
| Energy | Eₙ = E₁(Z²/n²); E₁ = -13.6 eV | |
| Spectral Lines | 1/λ = R(1/n₁² - 1/n₂²); R = 1.097×10⁷ m⁻¹ | |
| X-ray λ_min | λ_min = hc/(eV₀) = 12400/V₀ × 10⁻¹⁰ m | |
| Nuclear Radius | R = R₀A^(1/3); R₀ = 1.1×10⁻¹⁵ m | |
| Decay Law | N = N₀e^(-λt) | |
| Half Life | T₁/₂ = 0.693/λ | |
| Mean Life | T_avg = T₁/₂/0.693 |
Heat & Thermodynamics
| Kirchhoff's Law | Emissive power/Absorptive power = E_blackbody | |
| Conduction | dQ/dt = -KA(dT/dx) | |
| Newton's Cooling | dθ/dt ∝ (θ - θ₀) | |
| Thermal Resistance | R = L/(KA) | |
| F to C | F = 32 + (9/5)C | |
| C to K | K = C + 273.16 | |
| Ideal Gas | PV = nRT | |
| Van der Waals | (P + an²/V²)(V - nb) = nRT | |
| Linear | L = L₀(1 + αΔT) | |
| Areal | A = A₀(1 + βΔT) | |
| Volume | V = V₀(1 + γΔT) | |
| Relation | α/1 = β/2 = γ/3 | |
| Stefan-Boltzmann | u = σAT⁴ (black); u = eσAT⁴ (grey) | |
| σ value | σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ |
Kinetic Theory of Gases
| Boltzmann Const. | k_B = nR/N | |
| KE | KE = (3/2)nRT | |
| RMS Speed | v_rms = √(3RT/M) | |
| Mean Speed | v̄ = √(8RT/(πM)) | |
| Most Probable | vₚ = √(2RT/M) | |
| Speed Order | v_rms > v̄ > vₚ | |
| Pressure | P = (1/3)ρv²_rms | |
| Equipartition | K = ½k_BT per DOF; U = (f/2)nRT |
Chemistry Formulas
Physical Chemistry
Atomic Mass & Concentration
| Molarity (M) | M = w×1000/(M_solute × V_mL) | |
| Molality (m) | m = 1000w₁/(M₁w₂) | |
| Mole Fraction | x₂ = n/(n + N) | |
| % w/w | m_solute/m_solution × 100 | |
| % w/v | m_solute/V_mL × 100 | |
| RAM | mass of 1 atom / (1/12 × mass of ¹²C) | |
| Density | ρ = PM/(RT) | |
| Vapour Density | V.D. = M_gas/2 ⟹ M_gas = 2×V.D. | |
| Avg. Atomic Mass | Ā = (a₁x₁ + a₂x₂ + ...)/100 | |
| Avg. Molar Mass | M̄ = (n₁M₁ + n₂M₂ + ...)/(n₁ + n₂ + ...) | |
| Normality | N = equivalents/V; N = M × vf | |
| Dilution | N₁V₁ = N₂V₂ | |
| Equivalent Wt | E = Atomic weight / Valency | |
| Hardness (ppm) | m_CaCO₃/m_water × 10⁶ | |
| Mole fraction | x₂ = MM₁×10³/(ρ×10³ - MM₂) | |
| Molality | m = x₂×10³/(x₁M₁) | |
| Molarity | M = mρ×10³/(10³ + mM₂) |
Atomic Structure
| Photon Energy | E = hν = hc/λ | |
| Photoelectric Eq. | hν = hν₀ + ½mₑv² | |
| Angular Momentum | mvr = nh/(2π) | |
| Radius | rₙ = 0.529(n²/Z) Å | |
| Velocity | vₙ = 2.18×10⁶(Z/n) m/s | |
| Energy | Eₙ = -13.6(Z²/n²) eV | |
| Rydberg Eq. | 1/λ = RZ²(1/n₁² - 1/n₂²) | |
| Heisenberg | Δx·Δp ≥ h/(4π) | |
| De Broglie | λ = h/(mv) = h/p = h/√(2mK) | |
| Thermal λ | λ = h/√(2πmk_BT) | |
| Bohr Condition | 2πr = nλ | |
| Orbitals in Subshell | 2l + 1 | |
| Max Electrons | 2(2l + 1) | |
| Orbital Ang. Momentum | L = (h/2π)√(l(l+1)) |
Thermodynamics
| 1st Law | ΔU = q + w | |
| 2nd Law | ΔS_universe = ΔS_sys + ΔS_surr > 0 (spontaneous) | |
| 3rd Law | S - S₀ = k_B ln Ω | |
| Internal Energy | U = (f/2)nRT; ΔE = (f/2)nRΔT | |
| Cp | Cₚ = γR/(γ - 1) | |
| Cv | Cᵥ = R/(γ - 1) | |
| Specific Heat | S = Δq/(mΔT) | |
| Isothermal (rev) | W = -nRT ln(Vf/Vi) | |
| Isobaric | W = P(Vf - Vi) | |
| Adiabatic (rev) | W = nR(T₂-T₁)/(γ-1); T₂V₂^(γ-1) = T₁V₁^(γ-1) | |
| General Gas Law | P₁V₁/T₁ = P₂V₂/T₂ |
Enthalpy
| Enthalpy | H = U + pV | |
| Isobaric ΔH | ΔH = Cₚ(T₂ - T₁) | |
| Isothermal ΔH | ΔH = 0 | |
| Adiabatic ΔH | ΔH = Cₚ(T₂ - T₁) | |
| Reaction ΔH | ΔH_rxn = H_products - H_reactants | |
| Standard ΔH | ΔH°r = ΣvΔH°f(products) - ΣvΔH°f(reactants) | |
| Resonance Energy | ΔH°_res = ΔH°f(expt) - ΔH°f(calc) |
Entropy & Gibbs Energy
| System Entropy | ΔS = ∫dq_rev/T | |
| ΔS (T,V change) | ΔS = nCᵥ ln(T₂/T₁) + nR ln(V₂/V₁) | |
| ΔS (T,P change) | ΔS = nCₚ ln(T₂/T₁) - nR ln(P₂/P₁) | |
| Reaction ΔS | ΔS_rxn = ΣΔS_products - ΣΔS_reactants | |
| Gibbs Energy | G = H - TS | |
| Gibbs Eq. | ΔG = ΔH - TΔS |
Gaseous State
| Temp. Conversion | C/100 = (K-273)/100 = (F-32)/180 | |
| Boyle's Law | P₁V₁ = P₂V₂ | |
| Charles's Law | V₁/T₁ = V₂/T₂ | |
| Gay-Lussac's | P₁/T₁ = P₂/T₂ | |
| Ideal Gas | PV = nRT | |
| Dalton's Law | P_total = ΣPᵢ; Pᵢ = xᵢP_total | |
| Mixture M | M_mix = (n₁M₁ + n₂M₂ + ...)/(n₁ + n₂ + ...) | |
| Graham's Law | r₁/r₂ = √(M₂/M₁) = √(d₂/d₁) | |
| Van der Waals | (P + an²/V²)(V - nb) = nRT | |
| Critical Constants | Vc = 3b; Pc = a/(27b²); Tc = 8a/(27Rb) | |
| U_rms | √(3RT/M) | |
| Ū (mean) | √(8RT/(πM)) | |
| U_MPS | √(2RT/M) |
Chemical Equilibrium
| Kc | K = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ | |
| Kp | Kp = Kc(RT)^Δn = Kx·P^Δn | |
| ΔG° and K | ΔG° = -RT ln K = -2.303RT log K | |
| ΔG and Q | ΔG = ΔG° + 2.303RT log Q | |
| Van't Hoff Eq. | log(K₂/K₁) = (ΔH/2.303R)(1/T₁ - 1/T₂) | |
| Degree of Dissoc. | α = moles dissociated / initial moles |
Ionic Equilibrium
| Ostwald Dilution | Ka = Cα²/(1-α) ≈ Cα²; α = √(Ka/C) | |
| pH | pH = -log[H⁺] | |
| pOH | pOH = -log[OH⁻] | |
| Kw | [H⁺][OH⁻] = 10⁻¹⁴ | |
| pKa, pKb | pKa = -log Ka; pKb = -log Kb | |
| Acidic Buffer | pH = pKa + log([Salt]/[Acid]) | |
| Basic Buffer | pOH = pKb + log([Salt]/[Base]) | |
| Ksp | Ksp = xˣ · yʸ · s^(x+y) for MxAy | |
| Mixture [H⁺] | [H⁺] = (N₁V₁ + N₂V₂)/(V₁ + V₂) |
Electrochemistry
| Gibbs-EMF | ΔG = -nFE_cell; ΔG° = -nFE°_cell | |
| Nernst Eq. | E = E° - (0.0591/n) log Q | |
| At Equilibrium | E = 0; log K_eq = nE°/(0.0591) | |
| 1st Law | w = Zq = Zit | |
| 2nd Law | W₁/E₁ = W₂/E₂; W/E = itη/96500 | |
| Conductance | G = 1/R; K = 1/ρ | |
| Equiv. Conductivity | λ_E = K×10³/N | |
| Molar Conductivity | λ_m = K×10³/M | |
| Kohlrausch α | α = λᶜm/λ∞m |
Chemical Kinetics
| Zero Order Rate | [A] = [A]₀ - kt | |
| First Order Rate | k = (2.303/t) log([A]₀/[A]) | |
| Arrhenius Eq. | k = Ae^(-Ea/RT) |
Inorganic Chemistry
Bonding
| Bond Order | BO = ½(Nb - Na) | |
| Dipole Moment | μ = q × d |
Mathematics Formulas
Algebra
Quadratic Equations
| General Form | ax² + bx + c = 0 (a ≠ 0) | |
| Quadratic Formula | x = (-b ± √(b² - 4ac))/(2a) | |
| Sum of Roots (α+β) | -b/a | |
| Product of Roots (αβ) | c/a | |
| Discriminant | D = b² - 4ac | |
| From Roots | x² - (α+β)x + αβ = 0 | |
| Vertex x-coord | x = -b/(2a) | |
| Extreme Value | -D/(4a) | |
| Common Root | α = (c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁) | |
| Both Roots Common | a₁/a₂ = b₁/b₂ = c₁/c₂ | |
| General | ax² + 2hxy + by² + 2gx + 2fy + c = 0 | |
| Linear Factors Cond. | abc + 2fgh - af² - bg² - ch² = 0; h² - ab > 0 |
Sequence & Series
| nth Term | Tₙ = a + (n-1)d | |
| Sum Sₙ | n/2[2a + (n-1)d] = n/2(a + l) | |
| AP Condition | a,b,c in AP ⟹ 2b = a + c | |
| AP Mean | Aₖ = a + k(b-a)/(n+1); ΣAᵣ = nA | |
| nth Term | Tₙ = arⁿ⁻¹ | |
| Sum Sₙ | a(rⁿ - 1)/(r - 1) (r ≠ 1) | |
| Infinite GP | S∞ = a/(1-r) (|r| < 1) | |
| Harmonic Mean | H = 2ac/(a + c) | |
| AM ≥ GM ≥ HM | G² = AH | |
| Σr | n(n+1)/2 | |
| Σr² | n(n+1)(2n+1)/6 | |
| Σr³ | n²(n+1)²/4 |
Binomial Theorem
| General | (x+a)ⁿ = Σ C(n,r) xⁿ⁻ʳaʳ | |
| General Term | T(r+1) = C(n,r) xⁿ⁻ʳaʳ | |
| (1+x)ⁿ | Σ C(n,r) xʳ | |
| (1-x)ⁿ | Σ (-1)ʳ C(n,r) xʳ | |
| Middle (n even) | T(n/2 + 1) | |
| Middle (n odd) | T((n+1)/2) and T((n+3)/2) | |
| xᵐ occurs at r | r = (nα - m)/(α + β) | |
| Indep. of x | r = nα/(α + β) | |
| From end | Tᵣ(end) = T(n-r+2)(beginning) | |
| ΣCᵢ | C₀ + C₁ + ... + Cₙ = 2ⁿ | |
| Alternating Sum | C₀ - C₁ + C₂ - ... = 0 | |
| Even = Odd | C₀ + C₂ + ... = C₁ + C₃ + ... = 2ⁿ⁻¹ | |
| Sum of Squares | C₀² + C₁² + ... + Cₙ² = (2n)!/(n!)² | |
| (1+x)ⁿ (|x|<1) | 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ... | |
| Multinomial | (x₁+...+xₖ)ⁿ = Σ n!/(r₁!...rₖ!) x₁^r₁...xₖ^rₖ |
Coordinate Geometry
Straight Line
| Distance | d = √((x₁-x₂)² + (y₁-y₂)²) | |
| Section Formula | x = (mx₂ ± nx₁)/(m ± n) | |
| Slope | m = (y₁ - y₂)/(x₁ - x₂) | |
| Centroid | G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) | |
| Incentre | I = (ax₁+bx₂+cx₃)/(a+b+c), ... | |
| Area of Triangle | ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| | |
| Angle between Lines | tanθ = |m₁ - m₂|/|1 + m₁m₂| | |
| Angle Bisector | (ax+by+c)/√(a²+b²) = ±(a'x+b'y+c')/√(a'²+b'²) | |
| Collinearity | |x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1| = 0 | |
| Concurrency | |a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃| = 0 | |
| Distance | |ax₁ + by₁ + c|/√(a² + b²) | |
| Parallel Condition | a/a' = b/b' ≠ c/c' | |
| Perpendicular Cond. | aa' + bb' = 0 | |
| Parallel Distance | |C₁ - C₂|/√(a² + b²) | |
| Through Origin | ax² + 2hxy + by² = 0 | |
| Angle | tanθ = |2√(h² - ab)/(a + b)| |
Circle
| Standard | (x-a)² + (y-b)² = r² | |
| General | x² + y² + 2gx + 2fy + c = 0 | |
| Area | A = πr² | |
| Circumference | C = 2πr | |
| Diameter | d = 2r | |
| Parametric | x = h + rcosθ, y = k + rsinθ | |
| x-axis intercept | 2√(g² - c) | |
| y-axis intercept | 2√(f² - c) | |
| Slope form | y = mx ± a√(1 + m²) | |
| Point form | xx₁ + yy₁ = a² (T = 0) | |
| Parametric form | xcosα + ysinα = a | |
| Pair of Tangents | SS₁ = T² | |
| Tangent Length | L = √S₁ | |
| Director Circle | x² + y² = 2a² | |
| Orthogonality | 2g₁g₂ + 2f₁f₂ = c₁ + c₂ | |
| Radical Axis | S₁ - S₂ = 0 | |
| Family | S₁ + KS₂ = 0; S + KL = 0 | |
| Length | 2LR/√(R² + L²) | |
| Triangle Area | RL³/(R² + L²) | |
| Angle | tan∠ = 2RL/(L² - R²) |
Parabola
| Equation | y² = 4ax | |
| Focus | (a, 0) | |
| Directrix | x = -a | |
| Latus Rectum | 4a; Ends: L(a,2a), L'(a,-2a) | |
| Slope Tangent | y = mx + a/m (m ≠ 0) at (a/m², 2a/m) | |
| Normal at (x₁,y₁) | y - y₁ = -(y₁/2a)(x - x₁) | |
| Chord Length | (4/m²)√(a(1+m²)(a - mc)) | |
| Mid-point Chord | T = S₁ where S₁ = y₁² - 4ax₁ |
Ellipse
| Equation | x²/a² + y²/b² = 1; b² = a²(1 - e²) | |
| Eccentricity | e = √(1 - b²/a²), 0 < e < 1 | |
| Foci | S = (±ae, 0) | |
| Directrices | x = ±a/e | |
| Latus Rectum | 2b²/a = 2a(1 - e²) | |
| Parametric | x = acosθ, y = bsinθ | |
| Auxiliary Circle | x² + y² = a² | |
| Slope Tangent | y = mx ± √(a²m² + b²) | |
| Point Tangent | xx₁/a² + yy₁/b² = 1 | |
| Normal | a²x/x₁ - b²y/y₁ = a² - b² | |
| Director Circle | x² + y² = a² + b² |
Hyperbola
| Equation | x²/a² - y²/b² = 1; b² = a²(e² - 1) | |
| Foci | S = (±ae, 0) | |
| Directrices | x = ±a/e | |
| Latus Rectum | 2b²/a = 2a(e² - 1) | |
| Parametric | x = asecθ, y = btanθ | |
| Asymptotes | x/a ± y/b = 0 | |
| Slope Tangent | y = mx ± √(a²m² - b²) | |
| Point Tangent | xx₁/a² - yy₁/b² = 1 | |
| Normal | a²x/x₁ + b²y/y₁ = a² + b² = a²e² | |
| Eccentricity | e = √2 | |
| Vertices | (±c, ±c) | |
| Foci | (±√2c, ±√2c) | |
| Parametric | x = ct, y = c/t | |
| Tangent at P(x₁,y₁) | x/x₁ + y/y₁ = 2 | |
| Normal at P(t) | xt³ - yt = c(t⁴ - 1) |
Calculus
Application of Derivatives
| Tangent | y - y₁ = f'(x₁)(x - x₁) | |
| Normal | y - y₁ = -1/f'(x₁) · (x - x₁) | |
| Condition | f'(h) = (f(h) - b)/(h - a) | |
| Tangent | y - b = [(f(h)-b)/(h-a)](x - a) | |
| Rolle's Theorem | ∃ c∈(a,b): f'(c) = 0 [f(a) = f(b)] | |
| LMVT | ∃ c∈(a,b): f'(c) = (f(b)-f(a))/(b-a) | |
| Angle between Curves | tanθ = |m₁ - m₂|/|1 + m₁m₂| | |
| Cuboid Volume | V = lbh; SA = 2(lb + bh + hl) | |
| Cube | V = a³; SA = 6a² | |
| Cone | V = ⅓πr²h; CSA = πrl | |
| Cylinder | CSA = 2πrh; TSA = 2πr(h + r) | |
| Sphere | V = ⁴⁄₃πr³; SA = 4πr² | |
| Sector Area | A = ½r²θ |
Derivatives
| d/dx(xⁿ) | nxⁿ⁻¹ | |
| d/dx(ln x) | 1/x | |
| Product Rule | uv' + vu' |
Indefinite Integration
| Power Rule | ∫(ax+b)ⁿ dx = (ax+b)ⁿ⁺¹/[a(n+1)] + C | |
| Logarithmic | ∫dx/(ax+b) = (1/a)ln|ax+b| + C | |
| Exponential | ∫e^(ax+b) dx = (1/a)e^(ax+b) + C | |
| ∫sin(ax+b)dx | -(1/a)cos(ax+b) + C | |
| ∫cos(ax+b)dx | (1/a)sin(ax+b) + C | |
| ∫tan(ax+b)dx | (1/a)ln|sec(ax+b)| + C | |
| ∫sec x dx | ln|sec x + tan x| + C | |
| ∫csc x dx | ln|csc x - cot x| + C | |
| ∫dx/√(a²-x²) | sin⁻¹(x/a) + C | |
| ∫dx/(a²+x²) | (1/a)tan⁻¹(x/a) + C | |
| ∫dx/√(x²+a²) | ln|x + √(x²+a²)| + C | |
| ∫dx/√(x²-a²) | ln|x + √(x²-a²)| + C | |
| ∫dx/(a²-x²) | (1/2a)ln|(a+x)/(a-x)| + C | |
| ∫dx/(x²-a²) | (1/2a)ln|(x-a)/(x+a)| + C | |
| ∫√(a²-x²)dx | (x/2)√(a²-x²) + (a²/2)sin⁻¹(x/a) + C | |
| ∫√(x²+a²)dx | (x/2)√(x²+a²) + (a²/2)ln((x+√(x²+a²))/a) + C | |
| ∫√(x²-a²)dx | (x/2)√(x²-a²) - (a²/2)ln((x+√(x²-a²))/a) + C | |
| By Parts | ∫fg dx = f∫g dx - ∫[f'∫g dx]dx | |
| Substitution | If f(x) = t, then f'(x)dx = dt |
Definite Integration
| Fundamental Thm | ∫ₐᵇ f(x)dx = F(b) - F(a) | |
| Limit of Sum | ∫ₐᵇ f(x)dx = lim Σhf(a+rh); h=(b-a)/n | |
| Variable Change | ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx | |
| Symmetric (even) | ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx | |
| Symmetric (odd) | ∫₋ₐᵃ f(x)dx = 0 | |
| Periodic | ∫₀ⁿᵀ f(x)dx = n∫₀ᵀ f(x)dx | |
| Leibnitz | F'(x) = h'(x)f(h(x)) - g'(x)f(g(x)) | |
| Walli's (even) | ∫₀^(π/2) sinⁿx dx = [(n-1)!!/n!!]·(π/2) | |
| Walli's (odd) | ∫₀^(π/2) sinⁿx dx = (n-1)!!/n!! |
Trigonometry
Identities
| sin(A+B) | sinAcosB + cosAsinB | |
| cos(2A) | 2cos²A - 1 | |
| Sine Rule | a/sinA = b/sinB = c/sinC = 2R | |
| Cosine Rule | cosA = (b² + c² - a²)/2bc |
Inverse Trigonometric Functions
| sin⁻¹(-x) | -sin⁻¹x; Domain [-1,1]; Range [-π/2, π/2] | |
| cos⁻¹(-x) | π - cos⁻¹x; Domain [-1,1]; Range [0, π] | |
| tan⁻¹(-x) | -tan⁻¹x; Domain ℝ; Range (-π/2, π/2) | |
| cot⁻¹(-x) | π - cot⁻¹x; Domain ℝ; Range (0, π) | |
| sec⁻¹(-x) | π - sec⁻¹x; Domain |x|≥1 | |
| csc⁻¹(-x) | -csc⁻¹x; Domain |x|≥1 | |
| d/dx sin⁻¹x | 1/√(1-x²) | |
| d/dx cos⁻¹x | -1/√(1-x²) | |
| d/dx tan⁻¹x | 1/(1+x²) | |
| d/dx cot⁻¹x | -1/(1+x²) | |
| d/dx sec⁻¹x | -1/(|x|√(x²-1)) | |
| d/dx csc⁻¹x | 1/(|x|√(x²-1)) |
Vectors
Vectors
| Position Vector | AB⃗ = b⃗ - a⃗ | |
| Distance | AB = |a⃗ - b⃗| | |
| Section Formula | r⃗ = (na⃗ + mb⃗)/(m + n) | |
| Midpoint | (a⃗ + b⃗)/2 | |
| Definition | a⃗·b⃗ = |a⃗||b⃗|cosθ | |
| Component Form | a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃ | |
| Projection | a⃗·b⃗/|b⃗| | |
| Angle | φ = cos⁻¹(a⃗·b⃗/(|a⃗||b⃗|)) | |
| Perpendicularity | a⃗·b⃗ = 0 ⟺ a⃗ ⊥ b⃗ | |
| Definition | a⃗×b⃗ = |a⃗||b⃗|sinθ n̂ | |
| Parallelogram Area | |a⃗×b⃗| | |
| Unit Normal | n̂ = ±(a⃗×b⃗)/|a⃗×b⃗| | |
| Triangle Area | ½|a⃗×b⃗ + b⃗×c⃗ + c⃗×a⃗| | |
| Quadrilateral Area | ½|d⃗₁×d⃗₂| | |
| Lagrange's Identity | (a⃗×b⃗)² = |a⃗|²|b⃗|² - (a⃗·b⃗)² | |
| Scalar Triple | [a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) | |
| Cyclic Property | [a⃗ b⃗ c⃗] = [b⃗ c⃗ a⃗] = [c⃗ a⃗ b⃗] | |
| Coplanar | [a⃗ b⃗ c⃗] = 0 | |
| Tetrahedron Vol. | V = (1/6)|[a⃗ b⃗ c⃗]| | |
| Tetra Centroid | (a⃗+b⃗+c⃗+d⃗)/4 | |
| Vector Triple | a⃗×(b⃗×c⃗) = (a⃗·c⃗)b⃗ - (a⃗·b⃗)c⃗ |
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