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| """ | |
| Formula Sheet — JEE Advanced quick reference. | |
| /formulas [physics|chemistry|math] | |
| """ | |
| from typing import Optional | |
| FORMULAS = { | |
| "physics": { | |
| "Kinematics": [ | |
| "v = u + at", | |
| "s = ut + ½at²", | |
| "v² = u² + 2as", | |
| "s_n = u + a(2n−1)/2 (nth second)", | |
| "Range R = u²sin2θ/g", | |
| "H_max = u²sin²θ/(2g)", | |
| "Time of flight T = 2u sinθ/g", | |
| ], | |
| "Newton's Laws & Friction": [ | |
| "F = ma", | |
| "f_s ≤ μₛN (static)", | |
| "f_k = μₖN (kinetic)", | |
| "Impulse J = FΔt = Δp", | |
| ], | |
| "Work, Energy & Power": [ | |
| "W = F·d·cosθ", | |
| "KE = ½mv²", | |
| "PE (gravity) = mgh", | |
| "PE (spring) = ½kx²", | |
| "Power P = W/t = Fv", | |
| "Work-Energy theorem: W_net = ΔKE", | |
| ], | |
| "Circular & Gravitation": [ | |
| "F_c = mv²/r = mω²r", | |
| "F_g = Gm₁m₂/r² (G = 6.674×10⁻¹¹)", | |
| "g = GM/R²", | |
| "Escape velocity vₑ = √(2gR) = √(2GM/R)", | |
| "Orbital speed v = √(GM/r)", | |
| "Gravitational PE = −Gm₁m₂/r", | |
| ], | |
| "Rotational Motion": [ | |
| "τ = Iα = r × F", | |
| "L = Iω", | |
| "KE_rot = ½Iω²", | |
| "Solid sphere: I = 2/5 MR²", | |
| "Hollow sphere: I = 2/3 MR²", | |
| "Disk/cylinder: I = ½MR²", | |
| "Rod (centre): I = ML²/12", | |
| "Rod (end): I = ML²/3", | |
| ], | |
| "Oscillations (SHM)": [ | |
| "x = A sin(ωt + φ)", | |
| "ω = 2π/T = 2πf", | |
| "Simple pendulum: T = 2π√(L/g)", | |
| "Spring-mass: T = 2π√(m/k)", | |
| "v_max = Aω at equilibrium", | |
| "E_total = ½kA² = ½mω²A²", | |
| ], | |
| "Waves": [ | |
| "v = fλ", | |
| "Intensity ∝ A²", | |
| "Standing wave: λₙ = 2L/n", | |
| "Beat frequency = |f₁ − f₂|", | |
| ], | |
| "Thermodynamics": [ | |
| "Q = mcΔT (calorimetry)", | |
| "ΔU = Q − W (1st law)", | |
| "Isothermal: PV = const", | |
| "Adiabatic: PVᵞ = const (γ = Cp/Cv)", | |
| "η_Carnot = 1 − T_cold/T_hot", | |
| "PV = nRT (ideal gas, R = 8.314 J/mol·K)", | |
| ], | |
| "Electrostatics": [ | |
| "F = kq₁q₂/r² (k = 8.987×10⁹ N·m²/C²)", | |
| "E = kq/r² (point charge)", | |
| "V = kq/r", | |
| "C = Q/V (capacitance)", | |
| "Energy in capacitor = ½CV²", | |
| "Parallel plate: C = ε₀A/d", | |
| ], | |
| "Current Electricity": [ | |
| "V = IR (Ohm's law)", | |
| "P = IV = I²R = V²/R", | |
| "R_series = R₁ + R₂ + …", | |
| "1/R_parallel = 1/R₁ + 1/R₂ + …", | |
| "Kirchhoff KVL, KCL", | |
| ], | |
| "Magnetism": [ | |
| "F = qvB sinθ (Lorentz)", | |
| "F = BIL sinθ (force on wire)", | |
| "B (solenoid) = μ₀nI", | |
| "B (wire) = μ₀I/(2πr)", | |
| "Magnetic flux Φ = BA cosθ", | |
| "EMF = −dΦ/dt (Faraday)", | |
| ], | |
| "Modern Physics": [ | |
| "E = hf = hc/λ (h = 6.626×10⁻³⁴ J·s)", | |
| "E = mc²", | |
| "λ_de Broglie = h/(mv) = h/p", | |
| "KE_max = hf − φ (photoelectric)", | |
| "rₙ = n²a₀ (Bohr, a₀ = 0.529 Å)", | |
| "Eₙ = −13.6/n² eV (Hydrogen)", | |
| "t₁/₂ = ln2/λ = 0.693/λ", | |
| ], | |
| "Optics": [ | |
| "n₁ sinθ₁ = n₂ sinθ₂ (Snell's law)", | |
| "Mirror: 1/f = 1/v + 1/u", | |
| "Lens: 1/f = 1/v − 1/u", | |
| "Lens maker: 1/f = (n−1)(1/R₁ − 1/R₂)", | |
| "Resolving power ∝ 1/λ", | |
| ], | |
| }, | |
| "chemistry": { | |
| "Atomic Structure": [ | |
| "E_n = −13.6/n² eV (H atom)", | |
| "rₙ = 0.529n² Å (Bohr radius)", | |
| "λ = h/(mv) (de Broglie)", | |
| "Δx·Δp ≥ h/(4π) (Heisenberg)", | |
| ], | |
| "Moles & Stoichiometry": [ | |
| "n = m/M (moles)", | |
| "n = N/Nₐ (Nₐ = 6.022×10²³)", | |
| "n (STP) = V/22.4 L", | |
| "Molarity M = n/V(L)", | |
| "Normality N = M × n-factor", | |
| "% by mass = (solute/solution) × 100", | |
| ], | |
| "Gas Laws": [ | |
| "Boyle: P₁V₁ = P₂V₂ (const T)", | |
| "Charles: V₁/T₁ = V₂/T₂ (const P)", | |
| "Gay-Lussac: P₁/T₁ = P₂/T₂ (const V)", | |
| "Ideal: PV = nRT (R = 0.082 L·atm/mol·K = 8.314 J/mol·K)", | |
| "Graham: r₁/r₂ = √(M₂/M₁)", | |
| ], | |
| "Thermochemistry": [ | |
| "ΔH = H_products − H_reactants", | |
| "q = mcΔT (calorimetry)", | |
| "Hess's law: ΔH = ΣΔH_f(products) − ΣΔH_f(reactants)", | |
| "Bond energy: ΔH = Σ(bonds broken) − Σ(bonds formed)", | |
| ], | |
| "Chemical Equilibrium": [ | |
| "Kc = [products]/[reactants] (mol/L)", | |
| "Kp = Kc(RT)^Δn", | |
| "Le Chatelier's principle", | |
| "Q < K → forward ; Q > K → reverse", | |
| ], | |
| "Ionic Equilibrium (Acids/Bases)": [ | |
| "pH = −log[H⁺]", | |
| "pOH = −log[OH⁻]", | |
| "pH + pOH = 14 (25°C)", | |
| "Weak acid: [H⁺] ≈ √(Ka·C)", | |
| "Weak base: [OH⁻] ≈ √(Kb·C)", | |
| "Buffer (Henderson-Hasselbalch): pH = pKa + log([A⁻]/[HA])", | |
| "Kw = Ka × Kb = 10⁻¹⁴ (25°C)", | |
| ], | |
| "Electrochemistry": [ | |
| "E_cell = E_cathode − E_anode", | |
| "ΔG° = −nFE° (F = 96485 C/mol)", | |
| "Nernst: E = E° − (RT/nF)lnQ", | |
| "Faraday: m = (M × I × t)/(n × F)", | |
| ], | |
| "Kinetics": [ | |
| "Rate = k[A]^m[B]^n", | |
| "1st order: [A] = [A]₀e^(−kt)", | |
| "t₁/₂ (1st) = 0.693/k", | |
| "t₁/₂ (2nd) = 1/(k[A]₀)", | |
| "Arrhenius: k = Ae^(−Ea/RT)", | |
| ], | |
| "Coordination Chemistry": [ | |
| "CFT: Crystal Field Splitting Δ", | |
| "Strong-field: low spin ; Weak-field: high spin", | |
| "EAN rule = 18-electron rule", | |
| ], | |
| }, | |
| "math": { | |
| "Algebra": [ | |
| "Quadratic: x = (−b ± √(b²−4ac))/(2a)", | |
| "Sum of roots = −b/a ; Product = c/a", | |
| "AM ≥ GM ≥ HM", | |
| "Binomial: (a+b)ⁿ = ΣC(n,k)aⁿ⁻ᵏbᵏ", | |
| ], | |
| "Sequences & Series": [ | |
| "AP: aₙ = a + (n−1)d ; S = n(2a+(n−1)d)/2", | |
| "GP: aₙ = arⁿ⁻¹ ; S = a(1−rⁿ)/(1−r)", | |
| "S_∞ (GP, |r|<1) = a/(1−r)", | |
| "ΣnΣ = n(n+1)/2 ; Σn² = n(n+1)(2n+1)/6", | |
| ], | |
| "Trigonometry": [ | |
| "sin²θ + cos²θ = 1", | |
| "tan²θ + 1 = sec²θ", | |
| "1 + cot²θ = csc²θ", | |
| "sin(A±B) = sinA cosB ± cosA sinB", | |
| "cos(A±B) = cosA cosB ∓ sinA sinB", | |
| "Sine rule: a/sinA = b/sinB = c/sinC", | |
| "Cosine rule: a² = b² + c² − 2bc cosA", | |
| ], | |
| "Calculus — Differentiation": [ | |
| "d/dx(xⁿ) = nxⁿ⁻¹", | |
| "d/dx(eˣ) = eˣ", | |
| "d/dx(ln x) = 1/x", | |
| "Product rule: (uv)' = u'v + uv'", | |
| "Chain rule: d/dx[f(g(x))] = f'(g)·g'", | |
| "L'Hôpital: lim f/g = lim f'/g' (0/0 or ∞/∞)", | |
| ], | |
| "Calculus — Integration": [ | |
| "∫xⁿ dx = xⁿ⁺¹/(n+1) + C", | |
| "∫eˣ dx = eˣ + C", | |
| "∫sin x dx = −cos x + C", | |
| "∫cos x dx = sin x + C", | |
| "∫(1/x) dx = ln|x| + C", | |
| "Integration by parts: ∫uv' = uv − ∫u'v", | |
| ], | |
| "Vectors & 3-D": [ | |
| "Dot product: A·B = |A||B|cosθ", | |
| "Cross product: |A×B| = |A||B|sinθ", | |
| "Unit vector: â = A/|A|", | |
| "Section formula: P = (m·B + n·A)/(m+n)", | |
| ], | |
| "Probability": [ | |
| "P(A) = n(A)/n(S)", | |
| "P(A∪B) = P(A)+P(B)−P(A∩B)", | |
| "Bayes: P(A|B) = P(B|A)P(A)/P(B)", | |
| "Binomial: P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ", | |
| "Mean = np ; Variance = np(1−p)", | |
| ], | |
| "Matrices & Determinants": [ | |
| "det[[a,b],[c,d]] = ad − bc", | |
| "A⁻¹ = adj(A)/det(A)", | |
| "Cayley-Hamilton: A satisfies its characteristic eq.", | |
| "Rank ≤ min(m, n)", | |
| ], | |
| "Complex Numbers": [ | |
| "i² = −1 ; i³ = −i ; i⁴ = 1", | |
| "|z| = √(a²+b²)", | |
| "arg(z) = arctan(b/a)", | |
| "De Moivre: (cosθ+i sinθ)ⁿ = cos(nθ)+i sin(nθ)", | |
| ], | |
| }, | |
| } | |
| def get_formula_sheet(subject: Optional[str] = None) -> str: | |
| """Return formatted formula sheet for given subject or all subjects.""" | |
| subjects = ( | |
| [subject.lower()] if subject and subject.lower() in FORMULAS | |
| else list(FORMULAS.keys()) | |
| ) | |
| lines: list = [] | |
| for subj in subjects: | |
| data = FORMULAS[subj] | |
| lines.append(f"\n{'═'*60}") | |
| lines.append(f" {subj.upper()} — FORMULA SHEET") | |
| lines.append(f"{'═'*60}") | |
| for section, formulas in data.items(): | |
| lines.append(f"\n ── {section} ──") | |
| for f in formulas: | |
| lines.append(f" {f}") | |
| lines.append(f"\n{'═'*60}") | |
| lines.append(" Use: /formulas physics | /formulas chemistry | /formulas math") | |
| lines.append(f"{'═'*60}") | |
| return "\n".join(lines) | |