The Ultimate JEE Cheatsheet

─── Kinematics ───
  v = u + at
  s = ut + ½at²
  v² = u² + 2as
  s = ½(u + v)t
  For projectile (angle θ, velocity u):
    Range R = u²sin(2θ)/g
    Max Height H = u²sin²θ/(2g)
    Time of Flight T = 2u·sinθ/g

─── Newton's Laws ───
  F = ma (Second Law)
  Weight W = mg
  Friction f ≤ μN (static), f = μkN (kinetic)
  Circular motion: a_c = v²/r = ω²r, F_c = mv²/r
  Banking angle: tanθ = v²/(rg)

─── Work, Energy, Power ───
  W = F·d·cosθ = F·s (for constant force along displacement)
  KE = ½mv²
  PE (gravity) = mgh
  PE (spring) = ½kx²
  Power P = W/t = F·v
  Work-Energy Theorem: W_net = ΔKE = KE_final - KE_initial

─── Rotational Motion ───
  Torque τ = r × F = rFsinθ
  Moment of Inertia I = Σmr²
  Parallel axis theorem: I = I_cm + Md²
  Perpendicular axis: I_z = I_x + I_y
  Angular momentum L = Iω = mvr (for point mass)
  KE_rot = ½Iω²
  Rolling without slipping: v = ωR, a = αR
  For solid sphere: I = (2/5)mR²
  For hollow sphere: I = (2/3)mR²
  For solid cylinder: I = (1/2)mR²
  For ring: I = mR²

─── Gravitation ───
  F = Gm₁m₂/r²
  g = GM/R² (surface of earth)
  Orbital velocity v = √(GM/r)
  Escape velocity v_e = √(2GM/R) = √(2gR)
  Time period T = 2π√(r³/GM)
  Energy in orbit: E = -GMm/(2r)

─── Electrostatics ───
  Coulomb's Law: F = kq₁q₂/r² = (1/4πε₀)q₁q₂/r²
  Electric field E = F/q = kQ/r²
  Electric potential V = kQ/r = -∫E·dl
  Potential energy U = kQq/r
  Capacitance C = Q/V
  Parallel plate: C = ε₀A/d
  Energy stored: U = ½CV² = ½QV = Q²/(2C)

─── Current Electricity ───
  V = IR (Ohm's Law)
  Resistance R = ρL/A
  Series: R_eq = R₁ + R₂ + ...
  Parallel: 1/R_eq = 1/R₁ + 1/R₂ + ...
  Power P = VI = I²R = V²/R
  Kirchhoff's Laws:
    Junction rule: ΣI = 0 (current)
    Loop rule: ΣV = 0 (voltage)
  Wheatstone Bridge: P/Q = R/S (balanced condition)

─── Magnetism ───
  Force on moving charge: F = qv × B
  Force on wire: F = IL × B = BILsinθ
  Biot-Savart: dB = (μ₀/4π)(Idl × r̂)/r²
  Solenoid: B = μ₀nI
  Torque on loop: τ = NIABsinθ
  Magnetic moment M = NIA

─── Electromagnetic Induction ───
  Faraday's Law: EMF = -dΦ/dt
  Φ = B·A·cosθ (magnetic flux)
  Self-inductance: EMF = -L(dI/dt)
  Mutual inductance: EMF = -M(dI/dt)
  AC voltage: V = V₀sin(ωt)
  Impedance: Z = √(R² + (X_L - X_C)²)
  X_L = ωL, X_C = 1/(ωC)
  Resonance: ω = 1/√(LC), X_L = X_C

─── Wave Optics ───
  Young's Double Slit: y = nλD/d (bright fringes)
  Path difference: Δx = d·sinθ = nλ (bright), (n+½)λ (dark)
  Diffraction: Single slit width a, minima at a·sinθ = nλ
  Brewster's angle: tanθ_B = n₂/n₁
  Malus's Law: I = I₀cos²θ
  Lens formula: 1/v - 1/u = 1/f
  Mirror formula: 1/v + 1/u = 1/f
  Magnification: m = -v/u (mirror), m = v/u (lens)

─── Modern Physics ───
  Einstein's photoelectric effect: KE_max = hν - φ
  de Broglie wavelength: λ = h/(mv) = h/p
  Energy levels (Hydrogen): E_n = -13.6/n² eV
  Bohr radius: r_n = 0.529 × n² Å
  Mass-energy: E = mc²
  Radioactivity: N = N₀e^(-λt), t½ = 0.693/λ
  Nuclear binding energy: ΔE = Δm × c²

─── Thermodynamics ───
  First Law: ΔQ = ΔU + ΔW
  Isothermal: W = nRTln(V₂/V₁), ΔU = 0
  Adiabatic: W = (P₁V₁ - P₂V₂)/(γ-1), PV^γ = const
  γ = Cp/Cv = 5/3 (monatomic), 7/5 (diatomic)
  Carnot efficiency: η = 1 - T₂/T₁
  RMS speed: v_rms = √(3RT/M)
  Kinetic energy per molecule: (3/2)kT

─── Mole Concept & Stoichiometry ───
  Moles = mass/MW = Volume/L at STP = particles/NA
  At STP: 1 mole gas = 22.4 L
  NA (Avogadro) = 6.022 × 10²³
  Molarity M = moles/Volume(L)
  Molality m = moles/kg solvent
  Mole fraction X = n₁/(n₁ + n₂)

─── Thermodynamics ───
  ΔH = ΔU + Δn_gas × RT
  Hess's Law: ΔH_total = Σ ΔH (state function)
  Entropy: ΔS = Q_rev/T
  Gibbs: ΔG = ΔH - TΔS
  ΔG < 0: Spontaneous, ΔG = 0: Equilibrium

─── Chemical Kinetics ───
  Rate = k[A]^n[B]^m
  For first order: k = (2.303/t)log(a/(a-x))
  Half life: t½ = 0.693/k (first order)
  Arrhenius: k = Ae^(-Ea/RT)
  ln(k₂/k₁) = (Ea/R)(1/T₁ - 1/T₂)

─── Equilibrium ───
  Kp = Kc(RT)^Δn
  pH = -log[H⁺], pOH = -log[OH⁻]
  pH + pOH = 14 (at 25°C)
  Ka × Kb = Kw = 10⁻¹⁴
  Henderson equation: pH = pKa + log([A⁻]/[HA])

─── Electrochemistry ───
  E°cell = E°cathode - E°anode
  ΔG° = -nFE°cell
  Nernst: E = E° - (RT/nF)ln(Q)
  Faraday's laws: m = (M×I×t)/(n×F)
  Conductivity κ = 1/ρ = G × (l/A)
  Molar conductivity Λm = κ/c

─── Solutions ───
  Raoult's Law: P = X_A × P°_A (volatile solute)
  ΔTb = iKbm, ΔTf = iKfm
  Π = iCRT (osmotic pressure)
  Van't Hoff factor i: observed/calculated

─── Differentiation ───
  d/dx (xⁿ) = nx^(n-1)
  d/dx (sinx) = cosx, d/dx (cosx) = -sinx
  d/dx (tanx) = sec²x, d/dx (eˣ) = eˣ
  d/dx (lnx) = 1/x, d/dx (aˣ) = aˣln(a)
  Product rule: (uv)' = u'v + uv'
  Quotient rule: (u/v)' = (u'v - uv')/v²
  Chain rule: dy/dx = (dy/du)(du/dx)
  
  L'Hôpital's Rule: If 0/0 or ∞/∞ form, lim f(x)/g(x) = lim f'(x)/g'(x)

─── Integration ───
  ∫xⁿdx = x^(n+1)/(n+1) + C (n ≠ -1)
  ∫(1/x)dx = ln|x| + C
  ∫sinx dx = -cosx + C
  ∫cosx dx = sinx + C
  ∫sec²x dx = tanx + C
  ∫eˣdx = eˣ + C
  ∫aˣdx = aˣ/ln(a) + C
  
  Integration by parts: ∫u·dv = uv - ∫v·du (ILATE priority)
  Partial fractions for rational functions
  
─── Applications of Derivatives ───
  Increasing: f'(x) > 0, Decreasing: f'(x) < 0
  Maxima: f'(x) = 0, f''(x) < 0
  Minima: f'(x) = 0, f''(x) > 0
  Point of inflection: f''(x) = 0
  Tangent: y - y₁ = m(x - x₁), m = f'(x₁)
  Normal: y - y₁ = (-1/m)(x - x₁)
  Rate of change: dy/dt = (dy/dx)(dx/dt)

─── Differential Equations ───
  Order: highest derivative order
  Degree: power of highest order derivative (after removing fractions)
  Variable separable: f(x)dx = g(y)dy, integrate both sides
  Linear: dy/dx + Py = Q
    Solution: y × IF = ∫(Q × IF)dx, IF = e^(∫Pdx)

─── Quadratic Equations ───
  ax² + bx + c = 0
  Roots: α, β = (-b ± √(b²-4ac))/(2a)
  Sum = α + β = -b/a
  Product = αβ = c/a
  Discriminant D = b² - 4ac
  D > 0: Real distinct, D = 0: Equal, D < 0: Complex

─── Complex Numbers ───
  z = a + ib, |z| = √(a² + b²)
  Argument θ = tan⁻¹(b/a)
  Euler form: z = r(cosθ + isinθ) = re^(iθ)
  De Moivre: (cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ)
  Roots of unity: zⁿ = 1 → z = cos(2kπ/n) + isin(2kπ/n)

─── Matrices & Determinants ───
  det(AB) = det(A) × det(B)
  det(A⁻¹) = 1/det(A)
  det(Aᵀ) = det(A)
  Inverse: A⁻¹ = adj(A)/det(A)
  Rank: max number of linearly independent rows/columns

─── Sequences & Series ───
  AP: a, a+d, a+2d...; nth term = a+(n-1)d; Sum = n/2(2a+(n-1)d)
  GP: a, ar, ar²...; nth term = ar^(n-1); Sum = a(rⁿ-1)/(r-1)
  Infinite GP (|r|<1): S = a/(1-r)
  AM ≥ GM ≥ HM (for positive numbers)
  AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b)
  Binomial: (a+b)ⁿ = Σ C(n,r)a^(n-r)b^r
  Sum of series: use method of differences

─── Probability ───
  P(A∪B) = P(A) + P(B) - P(A∩B)
  Conditional: P(A|B) = P(A∩B)/P(B)
  Bayes: P(A|B) = P(B|A)P(A)/P(B)
  Independent: P(A∩B) = P(A)P(B)
  Binomial: P(X=r) = C(n,r)p^r(1-p)^(n-r)

─── Straight Line ───
  Distance: √((x₂-x₁)² + (y₂-y₁)²)
  Section: x = (mx₂+nx₁)/(m+n), y = (my₂+ny₁)/(m+n)
  Area of triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
  Slope m = (y₂-y₁)/(x₂-x₁) = tanθ
  Equation: y - y₁ = m(x - x₁)
  Parallel: m₁ = m₂, Perpendicular: m₁m₂ = -1
  Distance from point to line: |Ax₀+By₀+C|/√(A²+B²)

─── Circle ───
  Standard: x² + y² = r²
  Center (h,k): (x-h)² + (y-k)² = r²
  General: x² + y² + 2gx + 2fy + c = 0
  Center = (-g,-f), Radius = √(g²+f²-c)

─── Conic Sections ───
  Parabola: y² = 4ax (right opening), Focus = (a,0), Directrix = x = -a
  Ellipse: x²/a² + y²/b² = 1 (a>b), c² = a² - b², e = c/a < 1
  Hyperbola: x²/a² - y²/b² = 1, c² = a² + b², e = c/a > 1
  Eccentricity: e = distance from focus / distance from directrix