The user is preparing for JEE Mains 2027 and wants a comprehensive formula sheet with: 1. Entire formulas and reactions 2. Suitable diagrams 3. Useful data 4. Colorful text 5. Previous year questions (PYQs) and weightage of topics This is a massive request - JEE Mains covers Physics, Chemistry, and Mathematics with hundreds of formulas across ~25+ chapters. I need to be strategic here. Given the scope, I should create a comprehensive but well-organized guide. Since this is a text-based response, I can't literally use "colors" but I can use: - Markdown formatting (headers, bold, tables, emojis) - Unicode symbols and colored circle emojis (🔴🟢🔵🟡🟣 etc.) - ASCII/text diagrams where helpful - Tables for formulas - Weightage indicators Let me think about what to prioritize: 1. JEE Mains weightage by chapter (based on previous years' analysis) 2. Most important formulas per subject 3. Key chemistry reactions 4. Important constants/data 5. Exam pattern info The format should be extremely well-organized with: - Emojis for visual color coding - Tables for formula lists - Simple ASCII diagrams - Weightage percentages - Strategy tips Let me draft the structure: **PHYSICS** (high weightage chapters): 1. Current Electricity (~8%) 2. Electrostatics (~7%) 3. Modern Physics (~7-8%) 4. Ray Optics (~6-7%) 5. Rotational Motion (~6%) 6. Semiconductors (~5%) 7. Thermodynamics (~5%) 8. Magnetism (~5-6%) 9. Waves/Sound 10. Kinematics, Laws of Motion 11. Work Power Energy 12. Gravitation 13. SHM 14. AC/EMI 15. EM Waves 16. Communication (removed from new syllabus? Actually communication systems was removed in 2023-24 syllabus) Actually, for JEE 2025 syllabus, Communication Systems was removed, and some other topics trimmed. Let me recall the reduced syllabus: - Physics: Communication Systems removed; some topics like transistor configurations trimmed, EM waves applications trimmed - Chemistry: States of Matter, Surface Chemistry, s-block (hydrogen actually), polymers... wait let me recall. Chemistry removed: States of Matter, Surface Chemistry, s-Block Hydrogen (partial - hydrogen removed), Environmental Chemistry, Polymers, Chemistry in Everyday Life - Maths: Mathematical Induction, Mathematical Reasoning, and some parts of 3D geometry trimmed For JEE Main 2024/2025 syllabus (NTA): - Physics removed: Communication Systems, some experimental skills shifted... Actually experimental skills was added. Transistors removed? Let me think - the deleted topics in Physics: Communication systems; transistor action... Actually semiconductors logic gates remain. Physics deletions (2024 syllabus): Communication Systems, and topics within units like "transport in plants" no that's NEET. For JEE Main Physics: deleted = Communication systems, and some minor ones like "Lucas test"... no that's chemistry. Physics deleted: "Communication Systems" (Unit), "solid state electronics devices - transistor" partially? I recall logic gates stayed, transistors characteristic removed? Hmm. Chemistry deletions: States of Matter, Thomson/Rutherford models removed (kept in physics), Surface Chemistry, s-Block chemistry (Hydrogen), Environmental chemistry, Polymers, Chemistry in everyday life, general principles of metallurgy... wait no, Metallurgy was removed too? Let me recall: JEE Main 2024 Chemistry deleted chapters: States of Matter, Surface Chemistry, s-Block (Hydrogen), Environmental Chemistry, Polymers, Chemistry in Everyday Life. General Principles of Metallurgy - I believe it was also removed. Yes, "General Principles and Processes of Isolation of Metals" was deleted. Maths deletions: Mathematical Inductions, Mathematical Reasoning, and from 3D geometry a few topics... Actually Maths deleted: Mathematical induction, mathematical reasoning. Three dimensional geometry had some trimming? I think the 3D geometry unit merged with vectors and some parts removed like "equation of plane in different forms" — no wait. Let me be careful. Maths deletions 2024: Mathematical Induction, Mathematical Reasoning, and topics like "cube roots of unity"? Not sure. Also "triangle inequality"... I'll just state the main deletions: Mathematical Induction, Mathematical Reasoning. Since user is 2027 aspirant, current syllabus (2025) applies likely with minor tweaks. I'll base on latest known syllabus. Weightage data (approximate, based on PYQ analysis of recent years 2021-2024): PHYSICS: - Mechanics (Kinematics, NLM, WPE, Rotation, Gravitation): ~25-30% - Electrodynamics (Electrostatics, Current, Magnetism, EMI, AC): ~25% - Modern Physics + Semiconductors: ~12-15% - Optics: ~10% - Heat & Thermo: ~10% - Waves & SHM: ~8% - Units & Measurement + Errors: ~3% Chemistry: - Physical: ~30-35% - Organic: ~30-35% - Inorganic: ~30-35% High-weightage chapters: Chemistry: 1. Coordination Compounds (~8%) 2. Chemical Bonding (~8%) 3. GOC/Iso (~7%) 4. Thermo/Equilibrium (~7%) 5. Electrochemistry (~5-6%) 6. Chemical Kinetics (~5%) 7. p-block (~6%) 8. d&f block (~4%) 9. Atomic Structure (~4-5%) 10. Solutions, biomolecules, chemistry in everyday... (everyday life removed) Maths: 1. Calculus (~30%) 2. Coordinate Geometry (~15%) 3. Algebra (complex, quadratic, matrices, P&C, binomial, sequences): ~20% 4. Vectors & 3D: ~10% 5. Trigonometry: ~7% 6. Probability: ~5% Now, the challenge: "entire formulas and reactions" is impossible in one response — that would be a book. I should: 1. Provide a comprehensive high-yield formula sheet covering the most exam-relevant formulas 2. Organize by subject and chapter with weightage 3. Include key reactions (organic chemistry named reactions) 4. Include important data/constants 5. Use color emojis and formatting 6. Offer to expand specific chapters in follow-ups I should be honest that this is a curated high-yield version and offer to go deeper chapter by chapter. Format considerations: This will be a LONG response, which is appropriate given the request. I'll use: - Headers with emojis - Tables for formulas (Markdown tables render well) - Color emoji indicators for weightage (🔴 high, 🟡 medium, 🟢 low, 🟣 very high) - ASCII diagrams for key concepts - Constants table - Tips Let me draft the content: --- # 🎯 JEE MAINS 2027 MASTER FORMULA SHEET ## Exam Pattern Snapshot - 75 questions (25 per subject), 300 marks - Each subject: 20 MCQ + 5 Numeric - +4/-1 for MCQ, +4/0 for numeric ## Color legend 🟣 = Very High weightage (>3 questions) 🔴 = High (2-3 questions) 🟡 = Medium (1-2) 🟢 = Low (occasional) --- ## PHYSICS ### 1. Units, Dimensions & Errors 🟡 - Dimensional formulas table of key quantities - Error propagation: ΔZ rules ### 2. Kinematics 🟡 - v = u + at, s = ut + ½at², v² = u² + 2as - Projectile: R = u²sin2θ/g, H = u²sin²θ/2g, T = 2usinθ/g - Max range at 45°; complementary angles - Relative velocity ### 3. Laws of Motion 🟡 - F = ma, impulse = Δp - Friction: f = μN - Banking: tanθ = v²/rg - Pulley constraints ### 4. Work, Power, Energy 🟡 - W = F·d·cosθ, KE = ½mv², PE = mgh - Spring PE = ½kx² - Work-energy theorem - Power = F·v ### 5. Center of Mass & Collisions 🟡 - Elastic: v1 = ((m1-m2)/(m1+m2))u1 + ... - Loss in perfectly inelastic - Coefficient of restitution ### 6. Rotational Motion 🔴 - Moment of inertia table (ring, disc, rod, sphere, shell) - τ = Iα, L = Iω - Rolling: v = ωR, KE = ½mv²(1 + k²/R²) - Angular momentum conservation ### 7. Gravitation 🟢 - g' = g(1-2h/R), g' = g(1-d/R) - v_escape = √(2gR), v_orbital = √(GM/r) - T² ∝ r³, KE/PE/TE of satellite - Geostationary: T = 24h, h ≈ 36000 km ### 8. SHM 🔴 - x = A sin(ωt + φ) - T = 2π√(m/k), T = 2π√(l/g) - Velocity, acceleration max values - Simple pendulum, springs in series/parallel ### 9. Waves & Sound 🟡 - v = √(γP/ρ), v = √(T/μ) - Doppler effect formula - Beats, organ pipes - Standing waves harmonics ### 10. Thermal Physics 🔴 - Thermal expansion formulas - Q = mcΔT, Q = mL - Thermal resistance/conduction - Stefan's law, Wien's law, Newton's cooling ### 11. Thermodynamics (Physics) 🔴 - ΔU = Q - W - Isothermal: W = nRT ln(V2/V1) - Adiabatic: W = (P1V1 - P2V2)/(γ-1), PV^γ = const - Efficiency: η = 1 - T2/T1, COP ### 12. Kinetic Theory 🟡 - PV = nRT = ⅓Nmv_rms² - v_rms = √(3RT/M), speeds ratio - Degree of freedom, γ = (f+2)/f ### 13. Electrostatics 🔴 - Coulomb's law, E field formulas (point, ring, sheet, sphere) - Gauss's law applications - Dipole: p = qd, E_axial, E_equatorial, torque - Capacitors: series/parallel, energy, dielectrics - Sphere/dielectric formulas ### 14. Current Electricity 🔴 - Ohm's law, R = ρl/A - Series/parallel resistors - Kirchhoff, Wheatstone, potentiometer - Power, internal resistance - Drift velocity ### 15. Magnetism 🔴 - Force on charge: F = qvB sinθ, r = mv/qB, T = 2πm/qB - Force on wire, torque on loop - B formulas: wire, loop, solenoid - Bar magnet, tangent law, vibration magnetometer (removed?) - Moving coil galvanometer, ammeter shunt, voltmeter ### 16. EMI & AC 🟣 - ε = -dφ/dt, motional emf = Blv - Self/mutual inductance - Growth/decay: I = I0(1-e^(-t/τ)), τ = L/R - Reactances: XL = ωL, XC = 1/ωC - Impedance, resonance ω = 1/√(LC), power factor, rms values - Transformer ### 17. Ray Optics 🔴 - Mirror/lens formula, magnification - Power, combination - Refraction, TIR critical angle - Prism: A + δ = i + e, μ = sin((A+δm)/2)/sin(A/2) - Telescope/microscope magnification ### 18. Wave Optics 🟡 - YDSE: β = λD/d, fringe shift, thin film - Diffraction minima, polarization Brewster/Bragg... Bragg is in solid state? Actually Bragg's law removed? X-ray diffraction was removed in syllabus? I think "Bragg's law" may still be there... Actually wave optics deletions: "resolving power of microscope and telescope" removed? and "polarisation - plane polarized light, Brewster's law, uses" retained. Diffraction due to single slit retained. Brewster retained. ### 19. Modern Physics 🟣 - Photoelectric: hν = φ + KEmax - de Broglie λ = h/mv = h/√(2mKE) - Bohr model: r_n, E_n, v_n formulas - Hydrogen spectrum: 1/λ = R(1/n1² - 1/n2²) - Nuclear: mass defect, BE = Δm×931.5 MeV, radioactivity decay N = N0e^(-λt), half-life ### 20. Semiconductors 🔴 - Diode, LED, logic gates, Zener - Transistor removed from syllabus? Actually "transistor action" removed? Let me think... JEE Main 2024 Physics deleted: Communication Systems entire unit; from Semiconductors: Zener diode stayed? I believe logic gates stayed, transistor characteristics removed? Hmm, actually I recall semiconductors unit: "energy bands, conductors/insulators/semiconductors, intrinsic/extrinsic, p-n junction, diode, rectifier, LED, photodiode, solar cell, Zener, logic gates" retained; "transistor" removed? I'm not 100% sure. I think transistors were removed in the 2024 syllabus revision. Yes — "transistor action, transistor as amplifier" was removed. I'll include briefly but not emphasize. ### 21. EM Waves 🟡 - c = E/B = 1/√(μ0ε0), spectrum order, uses - Intensity = ½ε0E0²c, radiation pressure - Displacement current --- ## CHEMISTRY ### Physical 1. Mole concept 🟡: n = m/M, molarity, molality, mole fraction, % yield, empirical/molecular 2. Atomic Structure 🔴: Rydberg, de Broglie, Heisenberg, quantum numbers, E_n = -13.6Z²/n², radii 3. Chemical Bonding 🟣: VSEPR geometries table, hybridization, MOT bond order, dipole 4. Thermodynamics 🔴: ΔU, ΔH, Hess, ΔG = ΔH - TΔS, ΔG° = -nFE°... (that's electrochem), Cp-Cv = R 5. Equilibrium 🟡: Kp = Kc(RT)^Δn, pH, Ksp, buffer, solubility 6. Redox & Electrochemistry 🔴: Nernst, Faraday's laws, Λm = κ×1000/M, Kohlrausch 7. Kinetics 🔴: rate laws, first order t = (2.303/k)log(a/(a-x)), t½ = 0.693/k, Arrhenius 8. Solutions 🟡: Raoult, colligative properties 4 formulas, van't Hoff factor 9. Solid State 🟡 (deleted? No — solid state was NOT deleted from chemistry. States of matter (gaseous state) was deleted; solid state retained. Wait, actually let me recall. JEE Main 2024 chemistry deletions: States of Matter — yes deleted. Solid State — retained! Solid state is part of unit "States of Matter" originally... Hmm. NTA 2024 syllabus: Unit "States of Matter" deleted (which included gas laws, liquid state). "Solid State" was a separate unit and I believe it was retained... Actually no. Let me think carefully. JEE Main 2024 Chemistry deleted chapters (confirmed by coaching analyses): 1. States of Matter (gases and liquids) 2. Surface Chemistry 3. s-Block elements (Hydrogen part; s-block p-block? No...) — Actually: "Hydrogen" chapter deleted, s-block retained? Hmm. The deletion list commonly cited: States of Matter, Surface Chemistry, Hydrogen, s-Block? No wait. Let me recall from Allen/PW deletion lists for JEE Main 2024 Chemistry: - States of Matter ❌ - Hydrogen ❌ (s-block elements retained? Actually "s-block" is often taught with hydrogen; I recall "Hydrogen" deleted and "s-Block Elements" retained with group 1&2) - Environmental Chemistry ❌ - Polymers ❌ - Chemistry in Everyday Life ❌ - Surface Chemistry ❌ - General Principles of Metallurgy ❌ - Solid State — retained (still asked in 2024 papers) - Nuclear chemistry portion? Some trimmed. Hmm, I recall that in 2024 JEE Main, questions from Solid State did appear. Yes, solid state retained. And "p-block" got trimmed (group 13-14? No—p-block group 13 to 18 retained but ammonia... no). I'll go with: States of Matter, Surface Chemistry, Hydrogen, s-Block? Actually the official NTA deletion for chemistry: Unit 8 "States of Matter" removed? Hmm, official 2024 syllabus units: 1. Some Basic Concepts 2. Atomic Structure 3. Chemical Bonding 4. Chemical Thermodynamics 5. Solutions 6. Equilibrium 7. Redox & Electrochemistry 8. Chemical Kinetics 9. Classification & Periodicity 10. p-Block 11. d & f Block 12. Coordination Compounds 13. Purification & Characterisation of Organic Compounds 14. Some Basic Principles (GOC) 15. Hydrocarbons 16. Halogens (Haloalkanes/arenes) 17. Oxygen-containing (Alcohols, Phenols, Ethers, Aldehydes, Ketones, Carboxylic) 18. Nitrogen-containing (Amines) 19. Biomolecules 20. Principles of Practical Chemistry Deleted: States of Matter, Surface Chemistry, s-Block (merged/removed? I believe s-block deleted), Hydrogen, Environmental, Polymers, Everyday Life, Metallurgy. Yes! I now recall: the "s-Block elements" and "Hydrogen" were removed; p-block trimmed to groups 13-17? Something like that. Metallurgy removed. So chemistry units: no solid state? Wait, solid state — in the 2024 syllabus "States of Matter" unit included gas laws... Solid state... Hmm, I'm fairly certain Solid State was retained in JEE Main 2024 syllabus (questions appeared in Jan 2024 session from solid state — yes, there were questions on unit cells). Actually, hold on. Let me reconsider: NTA syllabus 2024 Chemistry Unit 5 was "Solutions"... The original units: 1. Some Basic Concepts, 2. Structure of Atom, 3. Classification of Elements & Periodicity, 4. Chemical Bonding, 5. States of Matter, 6. Thermodynamics, 7. Equilibrium, 8. Redox, 9. Hydrogen, 10. s-Block, 11. p-Block (13-14), 12. p-Block (15-18), 13. d & f, 14. Coordination, 15. Purification, 16. GOC, 17. Hydrocarbons, 18. Environmental, 19. Solid State, 20. Solutions, 21. Electrochemistry, 22. Kinetics, 23. Surface, 24. GOM (isolation of metals), 25. Haloalkanes, 26. Alcohols/Phenols/Ethers, 27. Aldehydes/Ketones/Acids, 28. Amines + diazonium, 29. Biomolecules, 30. Polymers, 31. Everyday Life. 2024 deletions: States of Matter, Hydrogen, s-Block, Environmental, Surface Chemistry, Polymers, Everyday Life, Isolation of Metals (GOM). Solid State — retained! Yes I'm now fairly confident solid state stayed (there were solid state questions in 2024 JEE Mains; coaching lists kept solid state). Hmm, but actually I also recall some saying solid state removed... Let me think about actual 2024 papers: Jan 2024 Shift questions included "unit cell edge length" — I believe yes, solid state was asked in 2024 and 2025. I'll include it as retained with low-medium weightage. To be safe I can note "check latest official syllabus." Hmm, to be safe, I'll present the high-yield content and add a note that a few chapters were trimmed in the 2024/2025 syllabus revision (list them) and advise verifying with the official NTA syllabus for 2027. ### Inorganic high-yield - Periodic trends 🟡 - Chemical bonding 🟣 (MOT) - p-block 🟡: key reactions (manufacture, oxyacids), anomalies - d & f block 🟡: colors, oxidation states, KMnO4/K2Cr2O7 - Coordination 🟣: IUPAC naming, VBT/CFT (Δo, CFSE), isomerism, EAN, magnetic moment √n(n+2), spectrochemical series - Qualitative salt analysis (practical chemistry) 🟢 ### Organic - GOC: inductive, resonance, hyperconjugation, acidity/basicity order, stability orders 🟣 - Isomerism: stereoisomers 2^n, R/S, E/Z - Named reactions list 🔴: Aldol, Cannizzaro, HVZ, Clemmensen, Wolff-Kishner, Friedel-Crafts, Reimer-Tiemann, Kolbe, Sandmeyer, Gattermann, Hofmann bromamide, Gabriel, Carbylamine, Coupling, Rosenmund, Stephen, Etard, Gatterman-Koch, Wurtz, Fittig, Finkelstein, Swarts, Riemer... etc. - Tests: Tollen, Fehling, iodoform, NaHCO3, Lucas - Conversion/reagents: LiAlH4, NaBH4, DIBAL-H, PCC, KMnO4, O3/Zn-H2O - Biomolecules 🟡: glucose reactions, peptides, vitamins - Practical organic 🟢 ## MATHEMATICS 1. Quadratic 🟡: nature of roots, sum/product, common roots, range 2. Complex numbers 🟡: i powers, modulus, argument, Euler, rotation, cube roots of unity (removed? "cube roots of unity" trimmed? I think kept), triangle inequality 3. Matrices & Determinants 🔴: properties, adjoint, inverse, rank, system of equations, characteristic equation, A² - (trA)A + det(A)I = 0 4. P&C 🔴: nPr, nCr, circular (n-1)!, derangements, distribution 5. Binomial 🟡: general term, middle term, greatest coefficient, sum of coefficients 6. Sequences & Series 🟡: AP/GP/HP sums, AGP, Σn, Σn², Σn³, AM-GM-HM 7. Trigonometry 🟡: identities, equations general solutions, properties of triangle (half-angle, sine/cosine rule) 8. Straight lines 🔴: forms, angle between lines, distance, family, bisectors? (Bisectors of angle between two lines trimmed? I think "shifting of origin, perpendicular distance, angle bisectors" — angle bisectors removed? Not sure. Keep basics.) 9. Circles 🟡: forms, tangent lengths, radical axis, family 10. Conics 🔴: parabola/ellipse/hyperbola parametric, tangent/normal, chord of contact, focal chord 11. Limits/Continuity/Differentiability 🟡 12. Differentiation & Applications (tangents, maxima, monotonic) 🔴 13. Integration 🔴: standard integrals table, by parts, definite properties 14. Differential equations 🟡: variable separable, linear, homogeneous 15. Vectors & 3D 🟣: dot/cross/triple products, line/plane forms, distance, angle, shortest distance 16. Probability 🔴: conditional, Bayes, binomial distribution 17. Statistics 🟡: mean, variance, SD Given the enormity, I need to decide how deep to go in one response. The user asked for "the entire formulas and reactions" — realistically impossible; I'll deliver a high-yield master sheet with the most crucial formulas per chapter, weightage tags, diagrams, constants, and named reactions, then offer chapter-wise deep dives. Format plan: - Title - How to use this sheet + weightage legend - Exam pattern table - PHYSICS section: chapter tables with formulas, weightage emoji, mini ASCII diagrams for select topics - CHEMISTRY section: formulas, named reactions table, tests, data - MATHS section: formula tables - Constants & Data tables (physics constants, math values, common organic data) - PYQ weightage summary tables (3 subjects) - Strategy & sources - Offer to expand any chapter I must keep tables compact. Use LaTeX? The rendering environment — markdown with inline math may not render LaTeX in all contexts. Safer to use plain text math (x = u + at style) with Unicode symbols (√, ², ³, θ, μ, Δ, π, ω, α, ∫). I'll use Unicode superscripts/subscripts where possible. LaTeX might not render; better to use Unicode plain math. That's what I'll do. Diagrams: ASCII sketches for: projectile parabola, wave, EM wave, standing wave nodes, logic gates table, SN2 backside attack, MOT energy diagram, YDSE, capacitor, prism. Keep them small and monospace in code blocks. Colored text: In markdown, I can't set actual colors, but I can use emojis (🔴🟠🟡🟢🔵🟣) to color-code. I'll explain the legend. Length: This will be long, but the request warrants it. I'll aim for comprehensive but scannable — heavy use of tables. Maybe 4000-6000 words equivalent. Let me be efficient — prioritize formula density over prose. Let me now write out the actual formulas carefully to ensure accuracy. PHYSICS KEY FORMULAS: Kinematics: - v = u + at; s = ut + ½at²; v² = u² + 2as - Snth = u + a/2(2n-1) - Projectile: T = 2u sinθ/g; H = u²sin²θ/(2g); R = u² sin2θ/g; Rmax = u²/g at 45° - Complementary angles give same range - Horizontal projectile from height h: t = √(2h/g), range = u√(2h/g) - Equation of trajectory: y = x tanθ − gx²/(2u²cos²θ) NLM: - F = dp/dt; Impulse J = Δp = FΔt - Friction: fs ≤ μsN, fk = μkN, angle of repose tanθ = μs - Banked road: v = √(rg tanθ); with friction: v²max = rg(μ+tanθ)/(1−μtanθ) - Conical pendulum: T = 2π√(L cosθ/g) - Vertical circle: vmin at top = √(gr), vmin at lowest for completing = √(5gr) WPE: - W = ∫F·dx; spring W = ½kx² - P = dW/dt = F·v - Power P = Fv COM & collisions: - vcm = Σmxi/Σmi - Elastic 1D: v1' = ((m1−m2)u1 + 2m2u2)/(m1+m2) - e = |v2'−v1'|/(u1−u2) - Perfectly inelastic common velocity v = (m1u1+m2u2)/(m1+m2); ΔKE = ½(m1m2/(m1+m2))(u1−u2)² - Oblique elastic equal masses: velocities at 90° Rotation: - MOI: ring mR²; disc ½mR²; rod center mL²/12, end mL²/3; shell ⅔mR²; solid sphere ⅖mR²; hollow cylinder? mR²; solid cylinder ½mR² - Perpendicular axis: Iz = Ix + Iy; parallel: I = Icm + Md² - Rolling KE = ½mv² + ½Iω² = ½mv²(1 + k²/R²) - a = g sinθ/(1 + k²/R²); friction f = (mg sinθ)(k²/R²)/(1+k²/R²) - τ = Iα, dL/dt = τ - Angular momentum conservation Gravitation: - F = GMm/r²; g varies: gh = gR²/(R+h)² ≈ g(1−2h/R); gd = g(1−d/R) - vesc = √(2GM/R) = 11.2 km/s - vorb = √(GM/r); TE = −GMm/2r; KE = GMm/2r; PE = −GMm/r - Kepler: T² = (4π²/GM)a³ - Gravitational potential V = −GM/r; field inside shell = 0 SHM: - x = A sin(ωt+φ); v = Aω cos(ωt+φ); a = −ω²x - vmax = Aω at mean; amax = Aω² at extremes - T = 2π√(m/k); springs: series 1/k = 1/k1+1/k2, parallel k = k1+k2 - Pendulum T = 2π√(l/g); in lift: T = 2π√(l/(g±a)) - Energy: E = ½mω²A²; KE = ½mω²(A²−x²); PE = ½mω²x² - T = 2π√(I/mgl) physical pendulum Waves: - v = √(E/ρ); string v = √(T/μ) - Sound: Newton-Laplace v = √(γP/ρ) = √(γRT/M); v ∝ √T; in air ~332 m/s at 0°C, +0.61 per °C - y = A sin(kx − ωt); v_wave = ω/k = fλ - Intensity ∝ A² - Doppler: f' = f(v ± vo)/(v ∓ vs) - Beats: fbeat = |f1−f2| - Organ pipes: closed fn = (2n−1)v/4L; open fn = nv/2L - End correction e = 0.6r - Wave speed transverse Thermal: - Linear α: l = l0(1+αΔT); areal β = 2α; volume γ = 3α - Q = mcΔT; Q = mL; calorimetry - Conduction: dQ/dt = kAΔT/l; thermal resistance l/kA; series/parallel - Radiation: P = εσAT⁴; net = εσA(T⁴−T0⁴); σ = 5.67×10⁻⁸ - Wien: λmT = 2.89×10⁻³ m·K - Newton cooling: dT/dt ∝ (T−T0) - Expansion of liquid in vessel: apparent γ = γliquid − γvessel KTG: - PV = nRT; R = 8.314; v_rms = √(3RT/M) = √(3P/ρ) - Speeds ratio v_mp : v_av : v_rms = 1 : 1.128 : 1.224 = √1 : √(8/π) : √3 - KE = (3/2)kT per molecule; U = (f/2)nRT - γ = 1 + 2/f; mono 5/3, dia 7/5, poly 4/3 - Mean free path λ = 1/(√2 πd²n) Thermo (physics): - ΔU = Q − W (chemistry convention) / ΔU = Q + W - Isothermal W = nRT ln(V₂/V₁) - Adiabatic: PV^γ = const; TV^(γ−1) = const; W = nR(T₁−T₂)/(γ−1) = (P₁V₁−P₂V₂)/(γ−1) - Isobaric W = PΔV = nRΔT - Isochoric W = 0 - Cycle: η = W/Qin; Carnot η = 1 − T₂/T₁; COP = Q₂/W = T₂/(T₁−T₂) - Efficiency of engine, refrigerator Electrostatics: - F = kq₁q₂/r², k = 9×10⁹ - E = F/q; point charge E = kq/r² - Ring: Eax = kqx/(x²+R²)^(3/2), max at x = R/√2 - Infinite wire: E = λ/2πε₀r = 2kλ/r - Sheet: E = σ/2ε₀; conductor surface: σ/ε₀ - Shell inside 0, outside kq/r², surface kq/R² - Solid sphere: Ein = kqr/R³ - Gauss: Φ = q_enc/ε₀; Φ = E·A - Dipole: p = q(2a); Eaxial = 2kp/r³; Eeq = kp/r³; τ = pE sinθ; U = −pE cosθ - Flux through Gaussian surface - Potential: V = kq/r; V = −∫E·dr - Capacitance C = Q/V; parallel plate C = ε₀A/d; with dielectric k: C = kε₀A/d - Series/parallel; Energy U = ½CV² = Q²/2C = ½QV - Energy density = ½ε₀E² - Spherical capacitor, cylindrical capacitor - Force between plates F = Q²/2ε₀A - Dielectric slab insertion with battery connected/disconnected — useful tips - Combination: sharing of charge, common V = total Q/total C Current electricity: - I = q/t = nAev_d; v_d = eEτ/m - R = ρl/A; ρ = m/(ne²τ) - Series R = ΣRi; parallel 1/R = Σ1/Ri - EMF: I = ε/(R+r); V = ε − Ir - Power P = VI = I²R = V²/R; max power transfer R = r - Cells: series nε, parallel ε/r... n cells: I = nε/(R+nr) etc. - Wheatstone balanced: P/Q = R/S - Meter bridge; potentiometer: ε₁/ε₂ = l₁/l₂; internal resistance r = R(l₁−l₂)/l₂ - Kirchhoff laws - Temperature dependence R = R₀(1+αΔT) Magnetism: - F = qv×B; circular motion r = mv/qB = √(2mK)/qB; T = 2πm/qB; pitch = v∥T - Helmholtz? rarely - F = Il×B (wire); two parallel wires F/l = μ₀I₁I₂/2πd; 1 definition ampere - Torque τ = NIAB sinθ; magnetic moment M = NIA - B_wire = μ₀I/2πd; B_arc center = μ₀Iφ/4πR; B_center loop = μ₀I/2R; solenoid B = μ₀nI; finite solenoid ends B/2 - Ampere's law ∮B·dl = μ₀Ienc - Bar magnet: Baxial = μ₀2M/4πr³? Wait Baxial = (μ₀/4π)(2M/r³); Beq = (μ₀/4π)(M/r³) - Magnet in field τ = MB sinθ; U = −MB cosθ - Magnetic materials: χ, μr = 1+χ - Cyclotron: v_max = qBR/m; f = qB/2πm (independent of v) - Moving coil galvanometer: I = (k/NAB)θ; current sensitivity - Ammeter shunt S = IgG/(I−Ig); voltmeter series resistance R = V/Ig − G EMI: - φ = BA cosθ; ε = −dφ/dt - Motional emf = Blv; force to move = B²l²v/R; power dissipated - Rotating rod: ε = ½Bωl²; rotating disc ε = ½BωR² - Rotating coil: ε = NBAω sinωt; εmax = NBAω - Self: ε = −L dI/dt; solenoid L = μ₀n²Al; L = Nφ/I - Mutual: ε₂ = −M dI₁/dt; M = μ₀n₁n₂Al - Growth: I = I₀(1−e^(−Rt/L)); decay I = I₀e^(−Rt/L); τ = L/R - Energy stored U = ½LI²; energy density = B²/2μ₀ - Charge flown q = Δφ/R = LΔI/R AC: - i = i₀sinωt; Irms = i₀/√2; avg over half cycle = 2i₀/π - XL = ωL; XC = 1/ωC; Z = √(R² + (XL−XC)²) - tanφ = (XL−XC)/R - Resonance ω₀ = 1/√(LC); Q factor = ω₀L/R = (1/R)√(L/C) - P = Vrms Irms cosφ; Pavg = (i₀²/2)R (only R consumes) - Wattless current - Series LCR at resonance: current max; parallel resonance current min (antiresonance) - Transformer: Vs/Vp = Ns/Np = Ip/Is; η - LC oscillations: f = 1/2π√(LC) Ray optics: - Mirror: 1/v + 1/u = 1/f; m = −v/u; f = R/2 - Sign conventions - Refraction: n₁sin i = n₂ sin r; n = c/v; apparent depth = real/n - TIR: sin C = 1/n - Prism: μ = sin((A+δm)/2)/sin(A/2); thin prism δ = (μ−1)A - Lens: 1/v − 1/u = 1/f; m = v/u; lens maker 1/f = (n−1)(1/R₁ − 1/R₂) - Power P = 1/f(m); combination P = P₁+P₂; lenses in contact - Displacement method - Silvered lens: P = 2Plens + Pmirror - Microscope: M = (v₀/u₀)(1 + D/fe) normal adjust = D... M = (L/f₀)(D/fe) approx - Telescope: M = f₀/fe; length = f₀+fe - Defects? removed I think (myopia etc. removed from syllabus 2024? "refraction of light through a prism" retained; "scattering of light – blue color of sky" removed? I think optical instruments retained: microscope, astronomical telescope) Wave optics: - YDSE: β = λD/d; dark/bright y_n = nλD/d; y_dark = (2n−1)λD/2d - In medium: λ' = λ/n; fringe width β' = β/n - Slab insertion: shift = (μ−1)tD/d (n−1) - YDSE with oblique incidence? rare - Angular width θ = λ/d - Single slit: minima a sinθ = nλ; width of central max = 2λD/a - Brewster: μ = tan θp - Malus: I = I₀cos²θ; unpolarized through polarizer I = I₀/2 - Resolving power removed? RP of microscope/telescope removed in 2024? I believe "resolving power" removed; "width of central maximum" retained. Polaroid retained. - Validity of ray optics: Fresnel distance zf = a²/λ (retained? probably) - Doppler in light? rare - Interference in thin films? Not in JEE Mains generally Modern physics: - Photoelectric: hν = W₀ + ½mv²max; eV₀ = hν − W₀; threshold λ₀ = hc/W₀ - Slope of V₀ vs ν = h/e - de Broglie: λ = h/p = h/√(2mK) = h/√(2mqV); electron: λ = 12.27/√V Å - Davisson-Germer - Bohr: rₙ = 0.529 n²/Z Å; vₙ = 2.18×10⁶ Z/n m/s; Eₙ = −13.6 Z²/n² eV - 1/λ = RZ²(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹ - Number of spectral lines = n(n−1)/2 - Ionization energy, series (Lyman UV, Balmer visible...) - Nuclei: R = R₀A^(1/3), R₀ = 1.2 fm; ρ ≈ 2.3×10¹⁷ kg/m³ - BE = Δm×931.5 MeV/u; BE/A max at Fe-56 (~8.75 MeV) - Radioactive: N = N₀e^(−λt); t½ = ln2/λ = 0.693/λ; mean life τ = 1/λ = t½/0.693; A = λN - Activity units: 1 Ci = 3.7×10¹⁰ Bq - Mass energy E = mc²; Q value Semiconductors: - Energy bands; n = n... carrier concentration formulas rarely - n-type: pentavalent (P, As, Sb); p-type: trivalent (B, Al, In) - p-n junction: depletion, barrier potential; forward/reverse bias - Diode equation I = I₀(e^(eV/kT) −1) - Rectifier: half-wave efficiency 40.6%, full-wave 81.2%; ripple frequency: half f, full 2f - Zener: voltage regulator - LED, photodiode, solar cell - Logic gates: AND, OR, NOT, NAND, NOR, XOR, XNOR; De Morgan's; universal gates - Transistor removed (2024 syllabus): I'll skip or mention briefly. Actually hold on — was the transistor removed? In the 2024 NTA syllabus for JEE Main Physics, Unit-10 "Electronic Devices": "Energy bands... p-n junction, semiconductor diode, I-V characteristics, diode as rectifier; special purpose diodes: LED, photodiode, solar cell, Zener; logic gates (OR, AND, NOT, NAND, NOR)". Transistor deleted. Yes, transistor was deleted. Also "Zener diode and voltage regulator" retained. Good. EM waves: - c = 1/√(μ₀ε₀) = 3×10⁸ - E₀/B₀ = c; E leads B in phase - Intensity I = ½ε₀E₀²c = E²rms/cμ₀ - Momentum p = U/c; radiation pressure = I/c (absorbing), 2I/c (reflecting) - Spectrum: γ < X < UV < visible < IR < micro < radio - Wavelength ranges: visible 400-700 nm - Displacement current id = ε₀ dΦE/dt - Uses of each wave type (PYQ asks this!) CHEMISTRY: Mole concept: - n = N/NA = m/M = V(STP)/22.4 L - Molarity M = n/V(L) = (10×d×%)/(M_m)... % by mass - Molality m = n solute/kg solvent - Mole fraction x_A = n_A/Σn - Dilution M₁V₁ = M₂V₂ - % yield, limiting reagent - Empirical formula Atomic structure: - E_photon = hν = hc/λ; λ = c/ν - Rydberg: ν̄ = R_H Z²(1/n₁²−1/n₂²) - de Broglie λ = h/mv; Heisenberg ΔxΔp ≥ h/4π - E_n = −13.6Z²/n² eV; r_n = 0.529n²/Z Å; v_n = 2.18×10⁶Z/n - Orbit angular momentum mvr = nh/2π; orbital: √(l(l+1))h/2π; spin s = 1/2 - Number of radial nodes = n−l−1; angular = l; total = n−1 - Quantum numbers; aufbau order; (n+l) rule - Effective nuclear charge, Slater rules rarely Chemical Bonding: - Lattice, octet rule, formal charge - Resonance - VSEPR shapes table: AB2 linear 180°, AB3 trig planar 120°, AB2E bent <120°, AB4 tetra 109.5°, AB3E trig pyramidal, AB2E2 bent 104.5°, AB5 trig bipyr 90/120, AB4E see-saw, AB3E2 T-shape, AB2E3 linear, AB6 octa 90°, AB5E square pyramidal, AB4E2 square planar, AB7 pentagonal bipyramidal... - Hybridization = ½(V + M − C + A) - MOT: bond order = (Nb−Na)/2; O₂ paramagnetic (2 unpaired); species orders - Dipole moment; % ionic character - Hydrogen bonding - MO configuration for N2, O2, CO, NO etc. and bond order table - Bond angle comparisons; back bonding - Fajan's rules; covalent character Thermodynamics: - ΔU = q + w; w = −PextΔV - ΔH = ΔU + ΔngRT - q = nCΔT; Cp − Cv = R; γ = Cp/Cv - ΔH = ΣΔHf(products) − ΣΔHf(reactants); Hess law - Bond enthalpy method - ΔS = qrev/T; ΔG = ΔH − TΔS; ΔG° = −RT lnK - Spontaneity conditions - Entropy changes in expansion: ΔS = nR ln(V₂/V₁) - Kirchhoff equation rare Equilibrium: - Kc, Kp = Kc(RT)^Δn - Le Chatelier - pH = −log[H+]; pH + pOH = 14; Ka·Kb = Kw - Weak acid: [H+] = √(KaC); α = √(Ka/C) (Ostwald) - Buffer: pH = pKa + log([salt]/[acid]); basic buffer pOH = pKb... - pKa + pKb = 14 - Salt hydrolysis pH formulas: strong acid+weak base pH = ½(pKw − pKb − logC)... useful - Ksp and solubility; common ion - Relation ΔG = ΔG° + RT lnQ Electrochemistry: - E°cell = E°cathode − E°anode - ΔG° = −nFE°cell; log K = nE°/0.059 - Nernst: E = E° − (0.059/n)log([products]/[reactants]) at 298K - pH of hydrogen electrode: E = −0.059 pH - Faraday: m = (ZIt); 1F = 96500 C per mole e⁻; charge Q = neF - Conductance: Λm = 1000κ/M; κ = (1/R)(l/A) - Kohlrausch: Λ°m = ν₊Λ°₊ + ν₋Λ°₋ - α = Λm/Λ°m; Ka = Cα²/(1−α) - Concentration cells - Electrolysis products, stoichiometry Kinetics: - Rate = −d[A]/dt = k[A]^m... - Zero order: [A] = [A]₀ − kt; t½ = [A]₀/2k - First order: k = (2.303/t)log([A]₀/[A]); t½ = 0.693/k - nth order: t½ ∝ 1/[A₀]^(n−1) - Arrhenius: k = Ae^(−Ea/RT); log(k₂/k₁) = (Ea/2.303R)(1/T₁ − 1/T₂) - Molecularity vs order - Rate constant units: (mol/L)^(1−n) s⁻¹ - Half life first order independent of concentration - Consecutive/side reactions rare; pseudo order Solutions: - Raoult: p_A = p°_A x_A; total P = p°A xA + p°B xB - Henry: p = KH x - Relative lowering: (p°−p)/p° = x_solute - Elevation: ΔTb = Kb·m·i; Depression: ΔTf = Kf·m·i - Osmotic: π = iCRT - van't Hoff factor i = observed/normal; degree of dissociation α = (i−1)/(n−1) - Azeotropes; ideal/non-ideal - Kb = RT²bM/1000ΔHvap... rarely needed Solid state: - 7 crystal systems, 14 Bravais (14 Bravais lattices number useful) - Unit cell: sc z=1, bcc z=2, fcc z=4; hcp z=6? (in unit cell basis for hexagonal: 6) - Packing efficiency: sc 52.4%, bcc 68%, fcc/hcp 74% - r relations: fcc r = a/2√2; bcc r = √3a/4; sc r = a/2 - Density ρ = ZM/(a³NA) - Ionic structures: NaCl 6:6, CsCl 8:8, ZnS 4:4, CaF2 8:4, rutile - Voids: tetrahedral 2n, octahedral n for n atoms - Radius ratio rules - Defects: Schottky, Frenkel; F-centers color - Piezo/ferro/antiferro - Close packing ABC=fcc, ABAB=hcp; coordination 12 Periodicity: - Trends: atomic radii, IE, EA, electronegativity - IE order exceptions: Be>B, N>O - EA: Cl>Br>F? Actually order: Cl > F > Br > I - Diagonal relationship - Effective nuclear charge p-block: - Key facts: inert pair effect; anomalous behavior - Group 13: BCl3 Lewis acid order; borax, boric acid, BF3; Al amphoteric; Tl+1 stable - Group 14: catenation C>Si; CO, CO2; silicates; SiO2 vs CO2 - Group 15: NH3 manufacture H2 + N2; HNO3 via Ostwald; P4 structure; oxyacids of P; NO, NO2 dimerization - Group 16: H2SO4 contact process; oxyacids of S; ozone - Group 17: halogens properties, bleaching powder; interhalogens - Group 18: XeF2, XeF4, XeF6 shapes - Common reactions with conditions & equations — need a compact table d & f: - Oxidation states, color (d-d transitions), magnetism μ = √n(n+2) BM - MnO4⁻ purple, Cr2O7²⁻ orange → Cr³⁺ green, Mn²⁺ pink/colorless - KMnO4 reactions (acidic/alkaline), K2Cr2O7 - Lanthanoid contraction; Ce⁴⁺; lanthanoid vs actinoid - Alloy/interstitial compounds Coordination: - Nomenclature rules - Werner; EAN = Z − ox state + 2CN - Isomerism: ionization, hydrate, linkage, coordination; geometric cis/trans; optical - VBT: inner/outer orbital; hybridizations: CN 4 dsp² square planar, sp³ tetra; 6: d²sp³ inner octa, sp³d² outer - CFT: Δo, d splitting, CFSE; strong field ligands CN⁻, CO; weak H2O, F⁻, Cl⁻ - Spectrochemical series: I⁻ < Br⁻ < SCN⁻ < Cl⁻ < F⁻ < OH⁻ < H2O < NH3 < en < NO2⁻ < CN⁻ ≈ CO - High/low spin: d⁴-d⁷ - Magnetic moment μ = √n(n+2) - Color, applications; chelation; EDTA - Stability constant GOC: - Carbocation stability: 3° > 2° > 1° > CH3; benzyl/allyl stabilized; resonance beats hyperconjugation - Carbanion reverse; radical similar to carbocation - Acidity: carboxylic > phenol > water > alcohol > alkyne > alkene/alkane - EWG increases acidity; EDG decreases - Basicity: amines in aq: 2° > 1° > 3° > NH3 (for alkyl); aniline less - Aromaticity: Huckel 4n+2 - Tautomerism; resonance rules - Inductive, hyperconjugation, electromeric - Reaction intermediates Hydrocarbons: - Alkane: Wurtz, Kolbe electrolysis, decarboxylation, cracking, isomerization, halogenation (free radical, order 3°>2°>1°) - Alkene: Markovnikov + anti-Mark (peroxide effect only HBr), ozonolysis, hydroboration, KMnO4 Baeyer test, polymerization - Alkyne: acidity, formation (dehydrohalogenation), hydration (Kucherov), Linlar/NH3-Na reduction to cis/trans - Benzene: EAS — nitration, sulfonation, halogenation, FC alkylation/acylation; directing effects; side chain oxidation - Aromaticity, Huckel rule; annulenes Haloalkanes: - SN1 vs SN2 table (rate, stereochemistry inversion, order, intermediate) - E1, E2, Saytzeff, Hofmann - Preparation: from alcohol (HCl/ZnCl2 Lucas, PX3, PX5, SOCl2), Sandmeyer, halogenation - Finkelstein NaI/acetone; Swarts AgF - Reactivity: RI > RBr > RCl; SN1: 3°>2°; SN2: 1°>3° - Allylic/benzylic fast Alcohols/Phenols/Ethers: - Preparation: hydration of alkene (Mark), hydroboration (anti-Mark), reduction of aldehyde/ketone/acids/esters, Grignard + aldehyde/ketone - Lucas test: 3° immediate turbidity, 2° 5 min, 1° no turbidity at RT - Dehydration order 3°>2°>1° (Saytzeff) - Oxidation: PCC 1°→aldehyde; KMnO4 → acid - Phenol: Reimer-Tiemann (salicylaldehyde), Kolbe-Schmidt (salicylic acid → aspirin), bromination (2,4,6-tribromophenol white ppt), nitration, coupling? azo with amines - Acidity of phenols; o/p directing - Ethers: Williamson synthesis, cleavage with HI (3° gives alkene via SN1 at high T? ether + HI), anisole reactions FC - Intramolecular dehydration at 413K vs 443K alkene Carbonyls: - Preparation: Rosenmund (acyl chloride → aldehyde, BaSO4), Stephen (nitrile → aldehyde, SnCl2/HCl), DIBAL-H, Etard (toluene → benzaldehyde, CrO2Cl2), Gatterman-Koch (CO+HCl, AlCl3/CuCl), ozonolysis, hydration of alkyne - Nucleophilic addition: HCN, NaHSO3, Grignard, NH2OH (oxime), 2,4-DNP, hydrazone - Aldol (α-H, dil NaOH), Cross-aldol, Cannizzaro (no α-H, conc NaOH) - Oxidation: Tollen (silver mirror), Fehling (red ppt — aliphatic), Benedict - Reduction: Clemmensen (Zn-Hg/HCl), Wolff-Kishner (N2H4/KOH) - Haloform: methyl ketones + I2/NaOH yellow CHI3; also ethanol, 2° alcohols with CH3CH(OH)- - Cannizzaro: 2HCHO + NaOH → CH3OH + HCOONa - Aldol: 2CH3CHO → β-hydroxy butanal - HVZ reaction for α-halogenation of acids - Carboxylic: acidity orders, HVZ, decarboxylation (soda lime), Hell-Volhard, esterification, acid chloride, anhydride, amide derivatives; reductions (LiAlH4 → alcohol, BH3 selective) - β-keto acid decarboxylation easy Amines: - Preparation: reduction of nitriles/nitro/amides; Gabriel phthalimide (1° only); Hofmann bromamide (amide + Br2/KOH → 1° amine, one C less); ammonolysis - Basicity order gas: 3°>2°>1°>NH3; aq depends solvation+steric - Hinsberg test; carbylamine test (1° amines, foul isocyanide) - Diazonium: sandmeyer (Cu2X2), Gattermann, coupling (azo dyes, pH), I2 easy - Nitrous acid reactions: 1° aliphatic → N2 + alcohol; 1° aromatic → diazonium; 2° → N-nitrosamine yellow oil; 3° → salt - Acylation, alkylation; quaternary salts; Hofmann elimination (least substituted alkene) - Aniline: bromination 2,4,6; protection via acylation (acetanilide p-product) - C6H5N2+Cl⁻ + H2O → phenol; + H3PO2 → benzene; + CuCN → benzonitrile Biomolecules: - Glucose: C6H12O6, aldohexose, open/pyranose, mutarotation, glycosidic - Reactions: with Br2 water (gluconic acid), HNO3 (saccharic), Tollen/Fehling positive (reducing) - Osazone with phenylhydrazine - Amino acids: 20, essential ones; zwitterion; peptide bond - Proteins: structures 1°2°3°4°, denaturation - Vitamins fat soluble A D E K, water B C; deficiency diseases table (scurvy C, rickets D, beriberi B1, pernicious anemia B12...) - Nucleic acids: DNA/RNA, base pairs A-T, G-C; nucleoside vs nucleotide - Carbohydrates classification; sucrose non-reducing; lactose; maltose; starch/cellulose - Enzymes; hormones Practical chemistry (Unit 20): tests — I listed above; group analysis; gas tests (turn lime water milky CO2, etc.) MATHS: Quadratic: - ax²+bx+c=0: roots (−b±√(b²−4ac))/2a; S = −b/a; P = c/a - D > 0 real distinct; =0 equal; <0 complex - Common roots conditions; range of quadratic: min at −b/2a (a>0): value (4ac−b²)/4a - Formation: x² − Sx + P - Graph-based conditions for roots positive/negative etc. (rarely now?) - Location of roots conditions Complex numbers: - z = x+iy; |z| = √(x²+y²); arg z = tan⁻¹(y/x) - Conjugate; Euler: e^iθ = cosθ + isinθ; e^(iπ) = −1 - (1+i)^n computations - Rotation: z' = z + (z2−z1)e^iθ - |z1+z2|² + |z1−z2|² = 2(|z1|²+|z2|²) - Cube roots of unity: 1, ω, ω²; 1+ω+ω² = 0; ω³ = 1 - Arg properties: arg(z1z2) = arg1+arg2 (mod 2π); |z1z2| = |z1||z2| - Triangle inequality - Amplitude problems, locus (circle, line, ellipse) Matrices & Determinants: - (AB)ᵀ = BᵀAᵀ; (AB)⁻¹ = B⁻¹A⁻¹; (AB)⁻¹ exists... - |adjA| = |A|^(n−1); A(adjA) = |A|I; adj(AB) = adjB·adjA - A⁻¹ = adjA/|A| - Skew-symmetric: Aᵀ = −A; diagonal of skew = 0; odd order skew det = 0 - A symmetric + skew = decomposition: A = ½(A+Aᵀ) + ½(A−Aᵀ) - Characteristic equation: λ² − (tr)λ + det = 0 for 2×2; Cayley-Hamilton - System: AX = B; |A|≠0 unique; |A| = 0 and (adjA)B = 0 infinite; ≠ 0 no solution - Cramer's rule; determinants properties row operations - Idempotent A²=A; nilpotent; involutory A²=I; orthogonal AᵀA=I P&C: - nPr = n!/(n−r)!; nCr = n!/r!(n−r)! - Circular: (n−1)!; necklace/garland: (n−1)!/2 - Σ nCr = 2^n; Σ nCr² = C(2n,n) - Distribution: identical/different; derangement !n = n!(1 − 1/1! + 1/2! − ...); !4 = 9, !5 = 44 - Division into groups; multinomial - Selection with repetition: C(n+r−1, r) - At least/at most; gaps method; grp method (men/women together) Binomial: - (a+b)^n: Tr+1 = nCr a^(n−r) b^r - Middle term: n even T(n/2+1); odd two middles - T of term from end = (n−r+2)th from beginning - Sum of coefficients: put x=1; alternating put x=−1 - Coefficient of x^r; term independent of x - Greatest binomial coefficient: (n+1)/2 terms - (1+x)^n expansions; approximate - Multinomial; binomial for any index rare now Sequences: - AP: an = a + (n−1)d; Sn = n/2(2a+(n−1)d) = n/2(a+l) - 3 terms a−d, a, a+d - GP: an = ar^(n−1); Sn = a(rⁿ−1)/(r−1); ∞: a/(1−r), |r|<1 - AGP: Sn = [a/(1−r)] + dr(1−r^(n−1))/(1−r)²... standard formula; infinite AGP: a/(1−r) + dr/(1−r)² - Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]² - AM ≥ GM ≥ HM; AM×HM = GM² - Relation between AM, GM, HM: if AM = A, GM = G, then roots of x² − 2Ax + G² - Harmonic progression nth term; sum special series method - Arithmetic-geometric series method Trigonometry: - sin²θ + cos²θ = 1; 1+tan² = sec²; 1+cot² = cosec² - sin(A±B); cos(A±B); tan(A±B) = (tanA±tanB)/(1∓tanAtanB) - 2A: sin2A = 2sinAcosA; cos2A = 1−2sin²A; tan2A = 2t/(1−t²) - 3A formulas - sinA ± sinB sums: 2 sin((A±B)/2)cos... - sin3θ = 3s − 4s³; cos3θ = 4c³ − 3c; tan3θ = (3t − t³)/(1 − 3t²) - General solutions: sinθ = 0 → nπ; sinθ = sinα → nπ + (−1)ⁿα; cosθ = cosα → 2nπ ± α; tanθ = tanα → nπ + α - Product to sum; max/min: amax sinθ + bcosθ = ±√(a²+b²) - Solution count in interval - Heights & distances formulas (angle of elevation, depression) — in JEE mains trigonometry? Heights & distances still in syllabus? I believe heights and distances was retained in trigonometry? Hmm, JEE Main 2024 syllabus trigonometry includes "heights and distances"? I'm not certain; older syllabus had it. I'll include briefly. - Properties of triangle: a/sinA = 2R; cosine rule; area = ½ab sinC = rs = √(s(s−a)(s−b)(s−c)); r = Δ/s; R = abc/4Δ Straight lines: - Slope m = (y₂−y₁)/(x₂−x₁) = tanθ - Forms: point-slope, two-point, slope-intercept, intercept, normal - Angle between lines: tanθ = |(m₁−m₂)/(1+m₁m₂)|; perpendicular m₁m₂ = −1 - Distance point-line: |ax₁+by₁+c|/√(a²+b²); distance between parallels - Foot of perpendicular, image - Concurrency; family of lines through intersection - Angle bisectors: (a1x+b1y+c1)/√(...) = ±(...) - Centroid, orthocenter, circumcenter, incenter coordinates - Area of triangle = ½|x₁(y₂−y₃)+...| - Position of point w.r.t. line Circles: - x²+y²+2gx+2fy+c = 0; center (−g,−f), r = √(g²+f²−c) - Tangent length = √S₁ - Tangent at (x₁,y₁) on circle: xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0 - Tangent from external in slope form: y = mx ± r√(1+m²) - Family: S + λS' = 0; circle on diameter: (x−x1)(x−x2)+(y−y1)(y−y2) = 0 - Orthogonal: 2g1g2 + 2f1f2 = c1+c2 - Common chord (radical axis): S − S' = 0 - Chord of contact: T = 0 - Director circle for circle same center? not needed - Parametric: x = h+rcosθ Parabola y² = 4ax: - Focus (a,0), directrix x = −a, LR = 4a - Parametric (at², 2at) - Focal chord, semi-latus rectum - Tangent: ty = x + at²; normal: y = −2at − at³ - Chord of contact T=0; pole rare Ellipse x²/a² + y²/b² = 1 (a>b): - e = √(1−b²/a²); foci (±ae, 0); LR = 2b²/a - Sum of focal distances = 2a - Tangent: xx₁/a² + yy₁/b² = 1; slope form y = mx ± √(a²m²+b²) - Auxiliary circle; eccentric angle - Parametric (acosθ, bsinθ) Hyperbola x²/a² − y²/b² = 1: - e = √(1+b²/a²); foci (±ae, 0); LR = 2b²/a - Conjugate hyperbola; rectangular e = √2 - Difference = 2a; asymptotes y = ±(b/a)x - Tangent slope form y = mx ± √(am² − b²) - Parametric (asecθ, btanθ) - Chord of contact Limits: - lim sinx/x = 1; (1+x)^(1/x) = e; (a^x−1)/x = lna; (e^x−1)/x = 1; ln(1+x)/x = 1 - L'Hôpital; expansion series (sin, cos, tan, e^x, ln(1+x), (1+x)^n) - lim x^n... form; 1^∞ form: lim f^g = e^(lim g(f−1)) - Squeeze; sandwich theorem - Trig limits forms Continuity/Differentiability: definitions; checking a, b parameters; greatest integer, |x| behavior; differentiability of composite, sum; LHD/RHD Differentiation: - Standard derivatives; chain, product, quotient - Parametric dy/dx = (dy/dt)/(dx/dt) - Logarithmic diff; implicit - Second derivative - Derivative of inverse functions - dy/dx of y^x, x^y, (sinx)^x type - Substitutions for inverse trig composite Applications: - Tangent: y−y₁ = m(x−x₁), m = dy/dx; normal slope = −1/m - Increasing: f'(x) > 0; decreasing < 0 - Maxima/minima: f' = 0 & sign change; second derivative test - Critical points; absolute extrema on [a,b]: check endpoints + criticals - Monotonic intervals; point of inflection f'' = 0 - Rate of change problems (ladder, cone water level) - Errors: dA/A ≈ ... (approximation) Integration: - ∫x^n dx = x^(n+1)/(n+1); ∫dx/x = lnx; ∫e^x = e^x; ∫a^x = a^x/lna - ∫sin = −cos; ∫cos = sin; ∫sec² = tan; ∫cosec² = −cot; ∫secx tanx = secx - ∫tan = ln|secx|; ∫cot = ln|sinx|; ∫sec = ln|secx+tanx|; ∫cosec = ln|cosecx−cotx| - ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a); ∫dx/√(a²−x²) = sin⁻¹(x/a); ∫dx/(x²−a²) = (1/2a)ln|(x−a)/(x+a)|; ∫dx/√(x²±a²) = ln|x+√(x²±a²)| - By parts ILATE; ∫e^x(f+ f') = e^x f + c - Special: ∫dx/(a+bcosx) etc. t = tan(x/2); ∫1/(asinx+bcosx) = 1/√(a²+b²) ln tan((x+α)/2) - Partial fractions table - Properties of definite: swap limits negative, add split, ∫₀^a f(x)dx = ∫₀^a f(a−x)dx (King's property), even/odd, periodic ∫ = sum over periods - Leibniz rule rarely - Area under curves: ∫(y_upper − y_lower)dx; between curves; parabola-line common problems; circle ellipse quarter DE: - Order/degree; variable separable; homogeneous: y = vx; linear: dy/dx + Py = Q, IF = e^∫Pdx, solution y·IF = ∫Q·IF dx - Exact? rarely; Bernoulli - Formation: d²y/dx² etc. - Population/growth/decay: dN/dt = λN - dy/dx = f(ax+by+c): substitution - Orthogonal trajectories rarely Vectors: - Position vector; section formula (internal m:n, external) - |a| = √(a·a); dot: a·b = |a||b|cosθ; projection of a on b = a·b/|b| - Cross: |a×b| = |a||b|sinθ; area of parallelogram = |a×b|; triangle = ½|a×b| - Coplanarity: [a b c] = 0; volume parallelepiped = |[abc]| - Scalar triple product identity [a×b·c] = [a b c]; (a×b)·c - Vector triple product: a×(b×c) = (a·c)b − (a·b)c - Unit vector along bisector: â1 + â2 normalized - Linear dependence: a×b + b×c + c×a = 0 coplanar - Moment of force M = r×F; work W = F·d 3D: - Direction cosines l²+m²+n² = 1; DRs - Line: r = a + λb; two point form; symmetric - Angle between lines cosθ = |b1·b2|/(|b1||b2|) - Plane: r·n̂ = d; general ax+by+cz = d; through 3 points; intercept form x/a+y/b+z/c = 1 - Normal form - Angle line-plane: sinθ = |b·n|/(|b||n|) - Angle between planes: cosθ - Distance point-plane: |ax₀+by₀+cz₀+d|/√(a²+b²+c²) - Distance point-line: |(a−b)×d|/|d|... |AP×b|/|b| - Skew lines shortest distance = |(a2−a1)·(b1×b2)|/|b1×b2| - Image of point in plane; foot of perpendicular - Family of planes; plane through line & point - Coplanar lines; intersection point - Distance between parallel planes Probability: - P(A∪B) = P(A)+P(B)−P(A∩B); mutually exclusive P(A∩B)=0 - Independent: P(A∩B) = P(A)P(B) - Conditional: P(A|B) = P(A∩B)/P(B) - Bayes: P(Ei|A) = P(Ei)P(A|Ei)/ΣP(Ej)P(A|Ej) - Total probability - Binomial: P(X=r) = nCr p^r q^(n−r); mean np, variance npq - At least one: 1 − (1−p)^n - Complement rules; odds - Bernoulli trials Statistics: - Mean = Σx/n; variance σ² = Σx²/n − (Σx/n)²; MD; σ for combined groups - Effect of change of origin/scale on mean, variance, SD - Median, mode relations; coefficient of variation = (σ/mean)×100 Constants/Data tables: Physics constants: c = 3×10⁸, h = 6.63×10⁻³⁴, ħ, e = 1.6×10⁻¹⁹, NA = 6.022×10²³, me = 9.1×10⁻³¹, mp = 1.67×10⁻²⁷, μ0 = 4π×10⁻⁷, ε0 = 8.85×10⁻¹², G = 6.67×10⁻¹¹, g = 9.8, R = 8.314, k = 1.38×10⁻²³, σ = 5.67×10⁻⁸, Wien 2.89×10⁻³, 1 eV = 1.6×10⁻¹⁹ J, hc = 1240 eV·nm (very useful!), amu = 931.5 MeV, Bohr radius 0.529 Å, Rydberg 1.097×10⁷, Earth data, 1 Cal = 4.18 J? (1 cal = 4.186 J), STP 22.4 L Chemistry data: Faraday 96500, Kw = 10⁻¹⁴, R values, 0.059 V, 8.314, ideal gas 22.4 L Math data: standard expansions, √2, √3 values; factorial values; binomial coefficient values; derivative/integral standard results PYQ weightage tables: I'll give approximate question counts per chapter per shift (JEE Main has ~25 Q per subject; weightages): Physics (out of ~25/shift, avg from 2021-2024): - Modern Physics (incl. Semiconductors): 3-4 Q - Electrostatics & Capacitance: 2-3 - Current Electricity: 2-3 - Magnetism & Moving Charges: 2-3 - EMI & AC: 2 - Ray Optics: 2 - Wave Optics: 1 - Thermodynamics & KTG: 2 - SHM & Waves: 2 - Rotation: 1-2 - NLM/Kinematics/WPE/COM: 3-4 combined - Gravitation: 0-1 - Units & Errors: 1 - EM Waves: 1 - Nuclei: included in modern Chemistry: - Coordination: 2 - Bonding: 2 - Organic named reactions/GOC: 3-4 - Physical numericals (Electrochem, Kinetics, Thermo, Equilibrium, Solutions): 5-6 total - p-block: 1-2 - Periodicity: 1 - Atomic structure: 1 - Mole: 1 - Biomolecules: 1 - Salt analysis/practical: 0-1 - d/f block: 1 - Amines/carbonyl: 2 Maths: - Calculus (limits to DE + area): 7-8 - Coordinate (lines, circles, conics): 3-4 - Vectors & 3D: 2-3 - Matrices/determinants: 2 - P&C + Binomial: 2 - Probability + statistics: 2 - Sequences: 1 - Quadratic/complex: 1-2 - Trigonometry: 1-2 - Functions/GT: 1 I should present these as approximate ranges and note variation by shift/year. Now, "colorful text": I'll use emoji color coding heavily: 🟣🟡🔴🟢 for weightage, 📘📗📙 for subjects, ⭐ for star formulas, ⚡ tricks, ⚠️ common mistakes, 💡 tips. Diagrams (ASCII): 1. Projectile 2. SHM spring 3. Standing wave 4. EM wave 5. YDSE 6. Logic gates 7. MOT energy diagram 8. CFT splitting 9. SN2 attack (backside) 10. Crystal structures? maybe skip 11. Lens/mirror ray? maybe small 12. Hyperbola/ellipse sketch? maybe skip to save space I'll include a handful of compact diagrams. Given the huge scope, I'll structure the response with clear headers and keep each chapter's formulas to the highest-yield ones. I'll also add a "how to use" note and end with an offer to do deep dives per chapter. One more consideration: The user said "add some useful data to use it in a proper way" — I interpret as important constants/values tables and shortcuts. And "according to previous questions and its weightage of this certain topic" — PYQ-based weightage tagging. I'll add a small PYQ example or "PYQ frequency" note per subject rather than full questions (space). Maybe add "Most repeated PYQ concepts" lists. Let me also double check a few high-risk formulas for accuracy: - Motional emf rotating rod: ε = ½Bωl² ✓ - Cyclotron frequency f = qB/(2πm) ✓ - Force between plates F = Q²/(2ε₀A) ✓ - Energy density E-field: ½ε₀E² ✓; B-field: B²/2μ₀ ✓ - Q factor = (1/R)√(L/C) = ω₀L/R ✓ - Average power LCR: Pavg = VrmsIrms cosφ; at resonance cosφ=1 ✓ - Transformer relation ✓ - Capacitance with dielectric slab thickness t: C = ε₀A/(d − t + t/k) ✓ - Sphere capacitance: C = 4πε₀R ✓ - Series capacitor formula ✓ - Combination of two spheres charge sharing Q₁/Q₂ = R₁/R₂ (same potential) ✓ - Drift velocity v_d = I/(neA) ✓ - Wheatstone ✓ potentiometer ✓ meter bridge ✓ - Post office box? rarely - Kirchhoff ✓ - Heating H = I²Rt = VIt; heat in resistor per time ✓ - Shunt ✓ - Spring balance? no - Wave: intensity ∝ A²/... ✓ - Organ pipe harmonics ✓ end correction 0.6r ✓ - Doppler sign conventions ✓ - Beats ✓ - Speed of transverse wave on string v = √(T/μ) ✓ - Newton's law of cooling approximation formula T = T₀ + (T_i−T₀)e^(−kt)? Keep qualitative + Stefan exact. - Efficiency heat engine ✓ - Coefficient of performance ✓ - Work done isothermal ✓ - Adiabatic work = (P₁V₁−P₂V₂)/(γ−1) ✓ = nR(T₁−T₂)/(γ−1) ✓ - Cp−Cv = R ✓ - v_rms = √(3RT/M) ✓ - g inside sphere linear ✓ - Escape velocity ✓ - Satellite energy −GMm/2r ✓ - Gravitational PE = −GMm/r ✓ - Kepler's third law T² = 4π²a³/GM ✓ - Gravitational field inside shell = 0 ✓ - Potential at center of ring = kQ/R... GM: V_ring = −GM/√(R²+x²) ✓ - Variation of g with rotation ω²R cosλ ✓ (rare) - SHM energy fractions ✓ - Physical pendulum ✓ - Collision formulas ✓ - Coefficient of restitution ✓ - Rocket equation? removed (v = u ln(m0/m)) — JEE Mains rarely; skip or include? Old syllabus had it; skip. - Banking with friction ✓ - Vertical circle ✓ - Projectile on incline? rare; skip. - Relative velocity of rain-man ✓ mention. - River-boat ✓ mention. Optics: - Displacement method: d² − D²/4d? For lens: x = √(D²−4fD)? The separation between two positions d = √(D²−4fD)? For object-screen distance D and two lens positions separated by d: f = (D²−d²)/(4D) ✓ - Combination of thin lenses ✓ - Silvered lens ✓ - Prism max at grazing? include thin prism ✓ - Microscope/telescope ✓ - Critical angle ✓ - Apparent shift ✓ - Lens cut problems (mention concept) - Brewster ✓ Malus ✓ - Fresnel distance a²/λ ✓ - Resolving power: removed in 2024 (verify) — I'll skip RP to be safe, or mention briefly. Actually "resolving power of microscope and telescope" appears deleted in 2024 syllabus. I'll skip it. Hmm, but I might include width of central max and Fresnel distance (retained). Modern: - e/m? Thompson — removed from syllabus? 2024 deleted "Thomson's model" & "Rutherford α-scattering"? Actually atomic structure in Physics 2024: "Alpha-particle scattering experiment; Rutherford's model; Bohr model..." retained; Thomson removed. Fine. - Davisson-Germer retained? "de Broglie... Davisson-Germer experiment" retained yes. - X-rays? "Continuous and characteristic X-rays" — I believe removed in 2024. Skip X-ray formulas. Hmm, dual nature unit: "Photoelectric effect, Hertz and Lenard's observations, Einstein's equation, particle nature, Davisson-Germer". X-rays removed. OK skip. - Nuclear fission/fusion retained ✓ - Mass defect ✓ - Radioactivity ✓ (alpha, beta decay equations, neutrino) - Binding energy curve ✓ Semiconductors retained topics as listed ✓ EM waves retained ✓ Maths: check some formulas: - Derangement values: !1=0, !2=1, !3=2, !4=9, !5=44 ✓ - Σ nCr² = (2n)!/(n!)² ✓ - AM-GM: for n numbers ✓ - Special: sum of AGP infinite = a/(1−r) + dr/(1−r)² ✓ - Cosine rule ✓; projection formula a = bcosC + ccosB ✓ - Area Δ = ½ab sinC; = abc/4R; = rs; Heron ✓ - Circumradius R = a/(2sinA) ✓ - Inradius r = Δ/s ✓; r = 4R sin(A/2)sin(B/2)sin(C/2) ✓ - Coordinates: distance formula; section; centroid; area of triangle ✓ - Angle bisector formula ✓ - Image of point ✓ formula - Distance parallel lines ✓ - Circle: tangent from external point length √S₁ ✓; condition tangency c² = a²(1+m²)... for y = mx + c tangent if c² = r²(1+m²) ✓ - Orthogonality condition ✓ - Parabola: parametric; focal distance = a + x₁ (for point x₁ on parabola); tangent ty = x + at²; normal y = −2at − at³; chord with mid point T = S₁ ✓ - Ellipse tangent; director circle x²+y² = a²+b² ✓ - Hyperbola: asymptotes; conjugate; director circle x²+y² = a²−b² (real if a>b) ✓ - Limit expansions ✓ - Derivatives standard ✓ including (uv)' etc. - Inverse trig derivative table ✓ - Max-min conditions ✓ - Rolle/LMV ✓ statements - Integrals table ✓ partial fractions ✓ - Definite integral properties: King's rule ✓ periodic ✓ even/odd ✓ - Walli's formula? rare but sometimes; skip or 1-line - Area ✓ - DE linear IF ✓ - Vectors: STP, VTP ✓ - Shortest distance skew ✓; parallel lines distance ✓ - Plane forms ✓ angle formulas ✓ point-plane distance ✓ image ✓ foot ✓ - Probability Bayes ✓ binomial mean/var ✓ - Statistics formulas ✓ combined variance formula: σ² = (n₁σ₁²+n₂σ₂²+n₁d₁²+n₂d₂²)/(n₁+n₂) ✓ Chemistry: check formulas: - Ostwald dilution law α = √(Ka/C) ✓ - Buffer Henderson ✓ - Salt pH: weak acid + strong base: pH = 7 + ½(pKa + logC) ✓; weak base + strong acid: pH = 7 − ½(pKb + logC) ✓ - Ksp ↔ solubility: AB: s = √Ksp; AB2: s = (Ksp/4)^(1/3); A2B3: s = (Ksp/108)^(1/5) ✓ - Δng ✓ Kp = Kc(RT)^Δn ✓ - Nernst ✓ relation with K: E° = 0.059/n log K ✓ - Λm = κ×1000/M ✓; α = Λm/Λ°m ✓; Ka = Cα²/(1−α) ✓ - Faraday m = (E×I×t)/(96500)... Z = E/96500 ✓ - ΔG° = −nFE° ✓; ΔG = ΔG° + 2.303RT logQ ✓ - Arrhenius two-temperature ✓ - Zero order t½ = [A]₀/2k ✓ - Colligative: ΔTb = i·Kb·m ✓; π = iCRT ✓; i formulas ✓ - Raoult with nonvolatile solute ✓ - Henry's law ✓ - Solid state density formula ✓; r-a relations ✓; voids ✓ - Bond order via MOT for homonuclear 2nd row: order up to N₂: σ2s < σ*2s < π2px = π2py < σ2pz; O₂, F₂: σ2s < σ*2s < σ2pz < π2px=π2py < π* ✓ - Hybridization formula H = ½(V+M−C+A) ✓ - EAN ✓; μ = √n(n+2) ✓ - CFSE: octa: t2g^x eg^y: CFSE = (−0.4x + 0.6y)Δo ✓ - Spectrochemical series ✓ - pH, Kw ✓ Organic named reactions table — make a table: Name | Reagents | Conversion | Key point. Keep ~25-30 most asked. Tests table: Tollen, Fehling, Iodoform, Lucas, Hinsberg, Carbylamine, NaHCO3, DNP, Ceric ammonium nitrate? (alcohol test), Bayer's KMnO4. Also add "Order & reactions reagents quick list": LiAlH4, NaBH4, DIBAL-H (ester → aldehyde), PCC, CrO2Cl2 (Etard), O3/Zn, hot KMnO4, cold dil KMnO4, NBS (allylic bromination), HBr/peroxide, Na/dry ether (Wurtz), RMgX... Also inorganic quick-reaction list: e.g., 4Fe + 10HNO3 dil → ..., NaCl + K2Cr2O7 + H2SO4 → chromyl chloride test, borax bead, brown ring test [Fe(H2O)5NO]²⁺, Nessler's reagent NH4+, K2MnO4 green, KMnO4 purple... Given the sheer volume, I'll compress into tables with minimal prose. It'll be a very long answer but that's what's asked. I'll add a note at the end offering chapter-wise expansion. Also add exam pattern & strategy section: - 75 Q, 180 min, 300 marks - Attempt order suggestion - Approx safe score targets for top NITs/IIITs (~ percentile 97+ → ~ top NIT CSE; JEE Mains score ~200+/300 for 99+ percentile... actually 99 percentile ≈ 185-220 varies by shift; 98.5+ percentile is cutoff-ish for NIT CSE general ~ around 97-98 percentile home state varies). Keep general advice. Also NTA syllabus deletion note for 2025 (apply to 2027, verify): Physics: Communication systems deleted; X-rays deleted; transistor deleted; resolving power deleted; etc. Chemistry: States of matter, surface chemistry, polymers, everyday life, environmental, hydrogen, s-block? (I need care — s-block: In the 2024 syllabus, "s-Block elements" as a standalone unit — hmm. Many coaching lists show "Hydrogen" deleted and "s-Block" deleted too. Actually I recall the official reduced chemistry syllabus 2024 units: ... Let me settle: The official unit list I wrote above (20 units) doesn't include s-block or hydrogen or states of matter or surface chemistry etc. p-block unit covers groups 13-18. Yes — s-block and hydrogen are NOT in the 2024 official syllabus. Confirmed by memory of unit list: Unit 9 was "p-Block Elements" covering group 13-18 after periodicity unit. Correct. Maths: Mathematical induction & mathematical reasoning deleted; also in 3D "equation of plane in different forms"? No — vectors and 3D retained fully; "scalar and vector triple product" retained; heights and distances? Trigonometry unit retained includes "heights and distances"? In the 2024 syllabus, "heights and distance" was retained under trigonometry? I believe yes, it's still there. Statistics retained. Bernoulli trials & binomial distribution retained (in probability). OK. I'll include a caution: "based on latest 2025 syllabus; verify NTA official for 2027." Now let me think about how to present "colorful": Since markdown can't do real colors in most renderers, I'll rely on emojis and bold. I'll explicitly define a legend: 🟣 VERY HIGH (4+ Q), 🔴 HIGH (2-3 Q), 🟡 MEDIUM (1-2 Q), 🟢 LOW (0-1 Q). Length management: This will be huge. I'll try to keep it tight: use compact tables, avoid explanations. Maybe ~600-900 lines. That's acceptable for "entire formulas" request, though I should still be reasonable. I'll prioritize and note that this is the condensed high-yield version, with offer to expand. Let me also decide on diagrams — small code blocks: 1. Projectile: ``` y | ___ | / \ | / \ R = u²sin2θ/g | /θ \ |/________________\___ x 45° max range ``` 2. SHM energy: ``` KE max |PE max ○————■————○ mean x extreme v max at mean, a max at extreme ``` 3. Standing wave: ``` N—A—N—A—N (N=node, A=antinode) 2 loops = 2nd harmonic ``` 4. EM wave: ``` E ↑ ↓ |\ /| propagation → B—+——+— E ⊥ B ⊥ direction ``` 5. YDSE: ``` S → ┌──────┐ ╱ │S1 ● ● ● ● fringe β = λD/d ╲ │S2 ``` 6. MOT diagram for O2/N2: ``` Energy ↑ σ*2p ─ ─ ─ ─ π*2p ─ ─ ↑↓ ↑ (O2: 2 unpaired) σ2p ↑↓ ↑↓ π2p ↑↓ ↑↓ ↑↓ ↑↓ σ*2s ↑↓ ↑↓ σ2s ↑↓ ↑↓ N₂ O₂ ``` 7. CFT: ``` eg ─ ─ (Δo) t2g ─ ─ ─ octahedral splitting ``` 8. SN2: ``` Nu⁻ δ− → [TS] → Nu—C + X⁻ ↖ attack from back X \ (Walden inversion) C ``` Maybe 4-6 diagrams to keep size manageable: projectile, standing wave/EM wave combined? I'll do: projectile, standing wave, EM wave, YDSE, MOT, CFT, SN2. That's 7 small ones. Also for maths maybe a conic diagram? skip — tables fine. Also add "Useful numeric data" — like sin/cos common values, powers of 2, factorial table, log values? Perhaps a small data page: - sin/cos/tan values at 0,30,45,60,90 - factorials up to 10 - 2^n up to 2^10 - √2 =1.414, √3=1.732, √5=2.236, π=3.14, e=2.72, ln2=0.693, ln10=2.303, 1/e = 0.37 - Conversions: 1 eV = 1.6×10⁻¹⁹ J; 1 amu = 931.5 MeV; hc = 12400 eVÅ; 1 MeV = 1.6×10⁻¹³ J; 1 dyne, 1 atm = 1.013×10⁵ Pa; R = 0.0821 L·atm; 0°C = 273K Chemistry: activity series? electrochemical series snippet; spectrochemical series; qualitative tests colors: Cr³⁺ green, Ni²⁺ green, Cu²⁺ blue, Fe³⁺ yellow/brown, Fe²⁺ light green, Co²⁺ pink, Mn²⁺ pink; flame colors: Na yellow, K violet, Ca brick red, Sr crimson, Ba apple green, Cu blue-green. Organic: pKa values? (rare for mains). Bond energies? skip. IUPAC priorities? brief. Functional group priority order for nomenclature: COOH > SO3H > COOR > COCl > CONH2 > CN > CHO > C=O > OH > NH2... include small. Named reaction table (organic) — must-have. ~30 entries, compact. Also "Most repeated PYQ topics" per subject bullet lists: Physics: logic gates truth table, Zener/regulator, transformer, AC power, potentiometer, meter bridge, YDSE fringe shift, microscope/telescope, photoelectric graphs, de Broglie, Bohr series, radioactivity half-life, CAP: dielectric slab battery scenarios, sphere charge sharing, Wheatstone, cell EMF internal resistance, magnetism loop/wire force, rotating rod emf, thermodynamics cycle P-V diagrams, radiation (Stefan/Wien), KTG speeds, SHM energy, waves (organ pipes, Doppler), rotation (rolling, MOI, angular momentum), gravitation (satellite energy, g variation), units dimensions errors, EM waves spectrum uses, projectiles, friction, vertical circles, elastic collisions. Chemistry: Nernst numericals, Λm Kohlrausch, pH buffers, Ksp, kinetics first-order, Arrhenius, colligative i, solid state density, VSEPR/MOT bond order, coordination isomerism/CFT color/magnetism, p-block group trends, d-block KMnO4, biomolecules vitamins, named reactions (aldol/cannizzaro/HVZ/sandmeyer/gabriel/hofmann), tests, GOC acidity order, isomerism counting, aromatic directing, polymer... (deleted) etc. Maths: determinant properties/adjoint, system solutions, P&C distribution, binomial coefficient sums, complex rotation/locus, straight line family + circle tangency, conic tangents/parametric, ellipse focal chord... limits expansion, definite integral properties + area, DE linear, 3D shortest distance/angle between planes, probability Bayes/binomial, statistics variance combined, vectors triple products, maxima-minima word problems, tangents/normal, monotonicity, function greatest integer... Strategy tips: NCERT for inorganic especially (NCERT line-based Qs), PYQ last 5 years, mock tests weekly, error notebook, formula revision cycle (D1/D3/D7/D21 spaced repetition), timing per section. Also add weightage summary tables per subject. I think I have a solid plan. Now write it. I'll be careful with Unicode math symbols. Use ² ³ ⁻ ¹ ⁰ subscripts where possible. Some formulas like x^(1/2) fine. Given constraints, final structure: 1. 🎯 Header + how to use + legend 2. 📋 Exam pattern table 3. 📚 Syllabus updates note (deleted chapters) 4. PHYSICS (with per-chapter weightage emoji + formula tables + diagrams) 5. CHEMISTRY (physical formulas, inorganic data, organic named reactions + tests + diagrams) 6. MATHEMATICS (formula tables) 7. 📊 Weightage summary tables (PYQ-based) 8. 🔢 Data/Constants tables 9. 💡 Strategy 10. Offer to expand Wait — order: maybe put weightage summary first for each subject then formulas. Better: integrate weightage emoji into each chapter header, plus a summary table per subject at the end of each subject or in strategy section. I'll integrate in headers and provide one consolidated summary table at the end. Let me be disciplined about table widths — keep 2-3 columns: Quantity/Concept | Formula. Sometimes add ⚡ note column. I'll write it now, keeping each chapter to its top formulas (10-20 rows max), full subject coverage. This is going to be long but the request demands it. I'll compress math chapters more (since tables of derivatives/integrals etc.). Also the user might benefit from knowing this is "Part 1 / high-yield" and they can ask for chapter deep-dives. End with that offer. Format check: Headers, tables, code blocks for diagrams, bold sparingly, emojis for color. Long response justified. No need for citations. Language: English (user's language). # 🎯 JEE MAINS 2027 — MASTER FORMULA + REACTION BANK ### *High-Yield Edition with PYQ Weightage, Diagrams & Exam Data* --- ## 📖 HOW TO USE THIS SHEET | Symbol | Meaning | Action | |--------|---------|--------| | 🟣 | VERY HIGH weightage (4+ Qs/shift) | Master 100% | | 🔴 | HIGH weightage (2–3 Qs/shift) | Master 90%+ | | 🟡 | MEDIUM (1–2 Qs/shift) | Strong grip needed | | 🟢 | LOW (0–1 Qs/shift) | Formula-level prep | **⭐ = Formula directly used in PYQs | ⚡ = Shortcut/Trick | ⚠️ = Common mistake zone** --- ## 📋 EXAM PATTERN SNAPSHOT | Feature | Details | |---------|---------| | Questions | 75 (25 per subject) | | Pattern | 20 MCQ + 5 Numerical per subject | | Marks | 300 (+4 correct, −1 MCQ, no negative in Numerical) | | Duration | 180 minutes | | Strategy | Physics ~45 min, Chemistry ~40 min, Maths ~60 min + 35 min buffer | --- # ⚛️ PART 1: PHYSICS ## 1. Units, Dimensions & Errors 🟡 | Quantity | Formula | |----------|---------| | % Error in product/quotient | ΔZ/Z = ΔA/A + ΔB/B | | % Error in power Z = AᵖBᑫ | ΔZ/Z = p(ΔA/A) + q(ΔB/B) | | Least count (Vernier) | LC = 1 MSD − 1 VSD | | Screw gauge | LC = Pitch/No. of divisions | ⭐ **Dimensions:** Force = [MLT⁻²], Energy = [ML²T⁻²], Power = [ML²T⁻³], Planck's h = [ML²T⁻¹] --- ## 2. Kinematics 🟡 | Concept | Formula | |---------|---------| | Equations of motion | v = u + at, s = ut + ½at², v² = u² + 2as | | Time of flight | T = 2u sinθ/g | | Max height | H = u²sin²θ/2g | | Range ⭐ | R = u²sin2θ/g; R_max = u²/g at θ = 45° | | Trajectory ⭐ | y = xtanθ − gx²/(2u²cos²θ) | | Horizontal projection | t = √(2h/g), Range = u√(2h/g) | ⚡ Same range at θ and (90° − θ) | ⚠️ Vertical component changes, horizontal stays constant ``` ___ H (apex, v_vertical = 0) / \ / θ \ /_____________\___ R ``` --- ## 3. Laws of Motion & Friction 🟡 | Concept | Formula | |---------|---------| | Impulse ⭐ | J = FΔt = Δp = m(v−u) | | Kinetic friction | f_k = μ_kN | | Angle of repose | tanθ = μ_s | | Banked road | v = √(rg·tanθ) | | Banked + friction ⭐ | v²_max = rg(μ+tanθ)/(1−μtanθ) | | Vertical circle ⭐ | v_top(min) = √(gr); v_bottom needed = √(5gr); T_bottom − T_top = 6mg | --- ## 4. Work, Power, Energy & Collisions 🟡 | Concept | Formula | |---------|---------| | Spring PE | U = ½kx² | | Work-Energy theorem | W_net = ΔKE | | Power | P = F·v | | Elastic collision (1D) ⭐ | v₁' = [(m₁−m₂)u₁ + 2m₂u₂]/(m₁+m₂) | | Perfectly inelastic | v = (m₁u₁+m₂u₂)/(m₁+m₂); ΔKE = ½·[m₁m₂/(m₁+m₂)](u₁−u₂)² | | Restitution | e = (v₂'−v₁')/(u₁−u₂) | ⚡ Equal masses elastic → velocities exchange | ⚡ Oblique elastic + equal mass → 90° between final velocities --- ## 5. Rotational Motion 🔴 | Body | MOI (about shown axis) | |------|----------| | Ring | mR² | | Disc | ½mR² | | Rod (centre/end) | mL²/12, mL²/3 | | Hollow sphere | ⅔mR² | | Solid sphere | ⅖mR² | | Solid cylinder | ½mR² | | Concept | Formula | |---------|---------| | Parallel axis ⭐ | I = I_cm + Md² | | Perpendicular axis | I_z = I_x + I_y (planar body) | | Rolling KE ⭐ | KE = ½mv²(1 + k²/R²) | | Rolling down incline ⭐ | a = gsinθ/(1 + k²/R²) | | Torque & L | τ = Iα, L = Iω, dL/dt = τ | ⚡ Order of rolling speed: Sphere > Disc > Ring | ⚡ Sphere reaches bottom first --- ## 6. Gravitation 🟢 | Concept | Formula | |---------|---------| | g at height ⭐ | g_h = g(1 − 2h/R) for h≪R | | g at depth | g_d = g(1 − d/R) | | Escape velocity | v_e = √(2gR) = 11.2 km/s | | Orbital velocity | v_o = √(GM/r); v_e = √2 · v_o | | Satellite energies ⭐ | KE = +GMm/2r, PE = −GMm/r, TE = −GMm/2r | | Kepler III | T² = 4π²r³/GM | ⭐ Geostationary satellite: T = 24 hr, h ≈ 36,000 km --- ## 7. SHM 🔴 | Concept | Formula | |---------|---------| | Displacement | x = A sin(ωt + φ) | | Velocity | v = ω√(A² − x²); v_max = Aω (at mean) | | Acceleration | a = −ω²x; a_max = Aω² (at extreme) | | Spring ⭐ | T = 2π√(m/k); Series: 1/k = 1/k₁+1/k₂ | | Pendulum | T = 2π√(l/g); In lift: g → (g±a) | | Total energy | E = ½mω²A² | ``` KE max ←———●———→ PE max mean extreme (v max) (a max, v=0) ``` --- ## 8. Waves & Sound 🟡 | Concept | Formula | |---------|---------| | String wave | v = √(T/μ) | | Sound in gas | v = √(γP/ρ) = √(γRT/M); v ∝ √T | | Beat frequency ⭐ | f_beat = |f₁ − f₂| | | Open pipe harmonics | fₙ = nv/2L | | Closed pipe ⭐ | fₙ = (2n−1)v/4L (odd only) | | End correction | e = 0.6r | | Doppler ⭐ | f' = f(v ± v_o)/(v ∓ v_s) | ⚠️ Doppler sign rule: *Observer moves toward → +v_o top; Source moves toward → −v_s bottom* ``` Standing wave: N—A—N—A—N (N = node/no motion, A = antinode/max motion) 2 loops between fixed ends = 2nd harmonic ``` --- ## 9. Thermal + KTG 🔴 | Concept | Formula | |---------|---------| | Thermal expansion | l = l₀(1+αΔT); γ = 3α, β = 2α | | Conduction ⭐ | H = kAΔT/l (thermal resistance = l/kA) | | Stefan's law | P = eσA(T⁴ − T₀⁴) | | Wien's law ⭐ | λ_m·T = 2.89×10⁻³ m·K | | rms speed ⭐ | v_rms = √(3RT/M) = √(3P/ρ) | | Speeds ratio | v_mp : v_avg : v_rms = 1 : 1.128 : 1.224 | | γ values | Mono = 5/3, Di = 7/5, Poly = 4/3; γ = 1 + 2/f | --- ## 10. Thermodynamics (Physics) 🔴 | Process | Work done | |---------|-----------| | Isothermal ⭐ | W = nRT ln(V₂/V₁) | | Adiabatic ⭐ | W = (P₁V₁ − P₂V₂)/(γ−1) | | Isobaric | W = nRΔT | | Isochoric | W = 0 | | Carnot efficiency ⭐ | η = 1 − T₂/T₁ | | Refrigerator COP | COP = T₂/(T₁−T₂) | ⚠️ First law: ΔU = Q − W (W = work done BY gas) — NTA follows this convention --- ## 11. Electrostatics & Capacitors 🔴 | Concept | Formula | |---------|---------| | Coulomb's law | F = kq₁q₂/r², k = 9×10⁹ Nm²/C² | | Infinite wire | E = 2kλ/r | | Infinite sheet | E = σ/2ε₀ | | Ring (axial) ⭐ | E = kqx/(x²+R²)^(3/2), max at x = R/√2 | | Solid sphere inside | E = kqr/R³ | | Dipole ⭐ | E_axial = 2kp/r³, E_equatorial = kp/r³; τ = pEsinθ | | Flux ⭐ | Φ = q_enc/ε₀ | | Parallel plate ⭐ | C = ε₀A/d; with slab: C = ε₀A/(d − t + t/k) | | Energy ⭐ | U = ½CV² = Q²/2C | | Energy density | u = ½ε₀E² | | Force between plates | F = Q²/2ε₀A | ⚠️ **Battery connected:** V constant, Q changes | **Battery disconnected:** Q constant, V changes --- ## 12. Current Electricity 🔴 | Concept | Formula | |---------|---------| | Drift velocity | v_d = I/neA | | Resistance | R = ρl/A; ρ = m/ne²τ | | Cell current | I = ε/(R + r); V = ε − Ir | | Max power transfer ⭐ | R_ext = r | | Wheatstone balance ⭐ | P/Q = R/S | | Meter bridge | S = R(100−l)/l | | Potentiometer ⭐ | ε₁/ε₂ = l₁/l₂; r = R(l₁−l₂)/l₂ | | Power | P = V²/R (parallel — compare), I²R (series — compare) | --- ## 13. Magnetism 🔴 | Concept | Formula | |---------|---------| | Charge in B ⭐ | r = mv/qB = √(2mK)/qB; T = 2πm/qB | | Helical path | Pitch = v_∥ × T | | Cyclotron ⭐ | f = qB/2πm (independent of speed!) | | Straight wire | B = μ₀I/2πd | | Loop centre | B = μ₀I/2R; Arc: B = μ₀Iθ/4πR | | Solenoid | B = μ₀nI | | Parallel wires ⭐ | F/l = μ₀I₁I₂/2πd | | Torque on loop | τ = NIABsinθ | | Bar magnet | B_axial = (μ₀/4π)(2M/r³) | | Ammeter shunt | S = I_gG/(I − I_g) | |---------------|---------------------| | Voltmeter | R = V/I_g − G | --- ## 14. EMI & AC 🟣 *(Guaranteed 2-3 Qs)* | Concept | Formula | |---------|---------| | Faraday's law | ε = −dΦ/dt | | Motional emf ⭐ | ε = Blv; Power lost = B²l²v/R | | Rotating rod ⭐ | ε = ½Bωl² | | Rotating coil | ε = NBAωsin(ωt) | | Inductance | L = μ₀n²Al (solenoid) | | Growth/decay | I = I₀(1−e^(−t/τ)), τ = L/R | | Stored energy | U = ½LI² | | Charge flown | q = LΔI/R | | Reactances ⭐ | X_L = ωL, X_C = 1/ωC | | Impedance | Z = √(R² + (X_L−X_C)²) | | Resonance ⭐ | ω₀ = 1/√(LC); I max, Z = R | | Power ⭐ | P_avg = V_rmsI_rms cosφ = I²_rmsR | | Q-factor | Q = ω₀L/R = (1/R)√(L/C) | | Transformer ⭐ | V_s/V_p = N_s/N_p = I_p/I_s | ``` EM Wave: E ↑ (E ⊥ B ⊥ direction) | ↘ B ——→ propagation → E₀/B₀ = c = 3×10⁸ m/s ``` --- ## 15. Ray Optics 🔴 | Concept | Formula | |---------|---------| | Mirror/Lens ⭐ | Mirror: 1/v + 1/u = 1/f; Lens: 1/v − 1/u = 1/f | | Lens maker ⭐ | 1/f = (n−1)(1/R₁ − 1/R₂) | | Power | P = 1/f(metres); Combination: P = P₁+P₂ | | Apparent depth | d_app = d_real/n; Shift = t(1 − 1/n) | | TIR ⭐ | sin C = 1/n | | Prism ⭐ | n = sin[(A+δm)/2]/sin(A/2); Thin: δ = (n−1)A | | Displacement method ⭐ | f = (D² − d²)/4D | | Microscope | M = (L/f₀)(D/f_e) | | Telescope | M = f₀/f_e; Length = f₀ + f_e | | Silvered lens ⭐ | P_eq = 2P_lens + P_mirror | --- ## 16. Wave Optics 🟡 | Concept | Formula | |---------|---------| | Fringe width ⭐ | β = λD/d | | nth bright/dark | yₙ = nλD/d; y = (2n−1)λD/2d | | Slab insertion ⭐ | Shift = (n−1)tD/d (fringes shift toward covered slit) | | Angular width | θ = λ/d | | Single slit minima | a sinθ = nλ; Central width = 2λD/a | | Brewster ⭐ | n = tanθ_p | | Malus' law | I = I₀cos²θ; Unpolarized → ½I₀ | ``` YDSE: S → [S₁] ● ● ● ● ● → Screen [S₂] β = λD/d ``` --- ## 17. Modern Physics 🟣 *(Highest ROI Chapter!)* | Concept | Formula | |---------|---------| | Photoelectric ⭐ | hν = φ + KE_max; eV₀ = hν − φ | | Graph slope | V₀ vs ν → slope = h/e | | de Broglie ⭐ | λ = h/√(2mK) = h/√(2mqV); Electron: λ = 12.27/√V Å | | Bohr radius ⭐ | rₙ = 0.529n²/Z Å | | Energy ⭐ | Eₙ = −13.6Z²/n² eV | | Velocity | vₙ = 2.18×10⁶ Z/n m/s | | Spectral line ⭐ | 1/λ = RZ²(1/n₁² − 1/n₂²) | | No. of lines | n(n−1)/2 | | Nuclear radius | R = 1.2×A^(1/3) fm | | Binding energy ⭐ | BE = Δm × 931.5 MeV | | Radioactive decay ⭐ | N = N₀e^(−λt); t½ = 0.693/λ; τ = 1/λ | ⚡ λ_electron = 12.27/√V Å (V in volts) — saves 1 min per question --- ## 18. Semiconductors 🔴 | Concept | Key Point | |---------|-----------| | n-type | Pentavalent dopant (P, As, Sb) | | p-type | Trivalent dopant (B, Al, In) | | Half-wave rectifier | η = 40.6%, ripple freq = f | | Full-wave rectifier | η = 81.2%, ripple freq = 2f | | Zener ⭐ | Operates in reverse bias; voltage regulator | | Gates | Output | |-------|--------| | AND | 1 only if both 1 | | OR | 0 only if both 0 | | NAND/NOR | Universal gates | | XOR | 1 if inputs differ | --- ## 19. EM Waves 🟡 | Concept | Formula | |---------|---------| | Speed | c = 1/√(μ₀ε₀); E₀ = cB₀ | | Intensity | I = ½ε₀E₀²c | | Radiation pressure | Absorbing: I/c; Reflecting: 2I/c | | Momentum | p = U/c | ⭐ **Spectrum order (PYQ favourite):** γ < X-ray < UV < Visible(400–700 nm) < IR < Microwave < Radio — *memorize the USES of each* (NTA asks directly!) --- # 🧪 PART 2: CHEMISTRY ## A. MOLE CONCEPT 🟡 | Concept | Formula | |---------|---------| | Mole | n = m/M = N/6.022×10²³ = V/22.4L (STP) | | Molarity ⭐ | M = (10 × d × %w/w)/M_molar | | Molality | m = moles solute/kg solvent | | Dilution | M₁V₁ = M₂V₂ | | Mixture molarity | M_mix = (M₁V₁ + M₂V₂)/(V₁+V₂) | --- ## B. ATOMIC STRUCTURE 🔴 | Concept | Formula | |---------|---------| | Energy of electron ⭐ | Eₙ = −13.6Z²/n² eV | | Bohr radius | rₙ = 0.529n²/Z Å | | Rydberg ⭐ | ν̄ = R_HZ²(1/n₁² − 1/n₂²) | | de Broglie | λ = h/mv | | Heisenberg ⭐ | Δx·Δp ≥ h/4π | | Nodes | Radial = n−l−1; Angular = l; Total = n−1 | | Orbitals | (n+l) rule for filling; max e⁻ in subshell = 2(2l+1) | --- ## C. CHEMICAL BONDING 🟣 *(2 Qs every shift)* | VSEPR | Shape | Angle | |-------|-------|-------| | AB₂ | Linear | 180° | | AB₃ | Trigonal planar | 120° | | AB₄ (CH₄) | Tetrahedral | 109.5° | | AB₅ (PCl₅) | Trigonal bipyramidal | 90°,120° | | AB₆ (SF₆) | Octahedral | 90° | | AB₂E₂ (H₂O) | Bent | 104.5° | | AB₄E (SF₄) | See-saw | — | | AB₃E₂ (ClF₃) | T-shape | — | **Hybridization ⭐:** H = ½(V + M − C + A) **MOT ⭐:** Bond order = (N_b − N_a)/2 - N₂ (14e⁻): BO = 3, diamagnetic - O₂ (16e⁻): BO = 2, **paramagnetic** (2 unpaired in π*) - O₂⁻: BO 1.5 | O₂⁺: BO 2.5 | O₂²⁻: BO 1 - **Stability order:** O₂⁺ > O₂ > O₂⁻ > O₂²⁻ ``` MOT Energy Diagram (up to N₂): σ2s < σ*2s < π2p = π2p < σ2p For O₂, F₂: σ2p comes BEFORE π2p ``` --- ## D. THERMODYNAMICS 🔴 | Concept | Formula | |---------|---------| | First law | ΔU = q + w; w = −P_extΔV | | Relation ⭐ | ΔH = ΔU + Δn_gRT | | Cp − Cv | = R | | Hess's law | ΔH = ΣΔH(products) − ΣΔH(reactants) | | Entropy of expansion | ΔS = nR ln(V₂/V₁) | | Gibbs energy ⭐ | ΔG = ΔH − TΔS; ΔG° = −RTlnK | | Spontaneity | ΔG<0 spontaneous; ΔG=0 equilibrium | ⚡ At equilibrium: ΔG° = −2.303RT logK --- ## E. EQUILIBRIUM 🟡 | Concept | Formula | |---------|---------| | Kp-Kc ⭐ | Kp = Kc(RT)^Δn | | pH | pH = −log[H⁺]; pH+pOH = 14 | | Weak acid ⭐ | [H⁺] = √(KaC); α = √(Ka/C) | | Acidic buffer ⭐ | pH = pKa + log([salt]/[acid]) | | pKa + pKb | = 14 | | Solubility ⭐ | AB: s = √Ksp; AB₂: s = (Ksp/4)^⅓; A₂B₃: s = (Ksp/108)^⅕ | --- ## F. ELECTROCHEMISTRY 🔴 | Concept | Formula | |---------|---------| | Nernst (298K) ⭐ | E = E° − (0.059/n)log([Red]/[Ox]) | | K from E° ⭐ | log K = nE°/0.059 | | ΔG° ⭐ | = −nFE°cell | | Faraday's law ⭐ | w = (E × I × t)/96500 | | Molar conductivity | Λm = 1000κ/M | | Kohlrausch ⭐ | Λ°m = ν₊Λ°₊ + ν₋Λ°₋ | | Degree of dissociation | α = Λm/Λ°m | --- ## G. KINETICS 🔴 | Concept | Formula | |---------|---------| | Zero order | [A] = [A]₀ − kt; t½ = [A]₀/2k | | First order ⭐ | k = (2.303/t)log([A]₀/[A]); t½ = 0.693/k | | Arrhenius ⭐ | log(k₂/k₁) = (Ea/2.303R)(1/T₁ − 1/T₂) | | Units of k | (mol/L)^(1−n) s⁻¹ | --- ## H. SOLUTIONS + COLLIGATIVE 🟡 | Property | Formula | |----------|---------| | Raoult | p_total = p°_A x_A + p°_B x_B | | ΔTb ⭐ | = i·Kb·m | | ΔTf ⭐ | = i·Kf·m | | Osmosis ⭐ | π = iCRT | | van't Hoff i | α = (i−1)/(n−1) | ⚡ Dissociation → i > 1; Association → i < 1 | NaCl i = 2, K₂SO₄ i = 3, benzoic acid (dimer) i = 0.5 --- ## I. SOLID STATE 🟡 | Structure | Z | Packing % | r vs a | |-----------|---|-----------|--------| | Simple cubic | 1 | 52.4% | r = a/2 | | BCC | 2 | 68% | r = √3a/4 | | FCC/HCP ⭐ | 4 | 74% | r = a/2√2 | **Density ⭐:** ρ = ZM/(a³N_A) ⭐ Voids per n atoms: Tetrahedral = 2n, Octahedral = n | NaCl 6:6, CsCl 8:8, ZnS 4:4 --- ## J. PERIODIC TABLE 🟡 ⭐ **Exceptions (asked repeatedly):** IE₁: Be > B, N > O | Electron affinity: Cl > F > Br > I | Atomic size: noble gases larger than halogens | IE₃ of Na ≫ IE₂ --- ## K. p-BLOCK 🟡 | Must-Know Reaction | Equation | |--------------------|----------| | Ammonia (Haber) | N₂ + 3H₂ ⇌ 2NH₃ (Fe cat., 700K, 200 atm) | | Ostwald ⭐ | 4NH₃ + 5O₂ → 4NO + 6H₂O (Pt, 500K) | | Contact process ⭐ | 2SO₂ + O₂ ⇌ 2SO₃ (V₂O₅) → H₂S₂O₇ → H₂SO₄ | | NO₂ dimer | 2NO₂ ⇌ N₂O₄ (colourless, low T) | | XeF₂/XeF₄/XeF₆ shapes | Linear/Square planar/Distorted octahedral | ⭐ Inert pair effect: Tl⁺, Pb²⁺, Bi³⁺ stable | Acidic oxide order: B₂O₃ < ... decreasing down group --- ## L. d & f BLOCK 🟡 | Concept | Data | |---------|------| | Magnetic moment ⭐ | μ = √[n(n+2)] BM | | Colours ⭐ | MnO₄⁻ purple, Cr₂O₇²⁻ orange, Cr³⁺ green, Mn²⁺ pink, Cu²⁺ blue | | Oxidation | KMnO₄ (Mn⁷⁺ → Mn²⁺ acidic, 5e⁻; Mn⁴⁺ neutral; Mn⁶⁺ alkaline) | | Lanthanoid contraction | Ce⁴⁺ good oxidant; caused by poor f-shielding | --- ## M. COORDINATION COMPOUNDS 🟣 *(Highest weightage in Inorganic!)* | Concept | Key Data | |---------|----------| | Magnetic moment | μ = √[n(n+2)] BM | | EAN | = Z − ox.state + 2×CN | | CFSE ⭐ | (−0.4x + 0.6y)Δ₀ for t2gˣegʸ | | Spectrochemical series ⭐ | I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < NO₂⁻ < CN⁻ < CO | | VBT hybridization | CN=4: dsp² (sq. planar), sp³ (tetra); CN=6: d²sp³ (inner), sp³d² (outer) | ``` CFT Splitting (Octahedral): ─ ─ eg (Δ₀ gap) ─ ─ ─ t2g ``` ⭐ Strong field (CN⁻, CO) → low spin → pairing | Weak field (H₂O, F⁻) → high spin ⚠️ **Isomerism counting** (ionisation, hydrate, linkage, geometrical cis/trans, optical) — direct PYQ every year! --- ## N. GOC + ISOMERISM 🟣 **Stability ⭐:** Carbocation: 3° > 2° > 1° > CH₃⁺ (resonance-stabilized > all) **Acidity ⭐:** RCOOH > phenol > H₂O > alcohol > HC≡CH **Basicity (aq.) ⭐:** 2° amine > 1° > 3° > NH₃ | Aniline < NH₃ (resonance) **Aromaticity ⭐:** Huckel rule (4n+2) π electrons + planar + cyclic --- ## O. NAMED REACTIONS 🟣 *(Master table — 1 Q guaranteed!)* | Reaction | Reagent | Conversion | |----------|---------|------------| | **Aldol** | dil. NaOH | Aldehyde/Ketone with **α-H** → β-hydroxy | | **Cannizzaro** | conc. NaOH | No α-H → alcohol + acid salt | | **Haloform** ⭐ | I₂/NaOH | CH₃CO– or CH₃CH(OH)– → yellow CHI₃ | | **Clemmensen** | Zn-Hg/HCl | C=O → CH₂ (acidic) | | **Wolff-Kishner** | N₂H₄/KOH | C=O → CH₂ (basic) | | **Rosenmund** | H₂/Pd-BaSO₄ | RCOCl → RCHO | | **Stephen** | SnCl₂/HCl | R–CN → RCHO | | **Etard** | CrO₂Cl₂ | Toluene → benzaldehyde | | **Reimer-Tiemann** ⭐ | CHCl₃/NaOH | Phenol → salicylaldehyde | | **Kolbe** ⭐ | CO₂/NaOH, H⁺ | Phenol → salicylic acid | | **Sandmeyer** ⭐ | Cu₂X₂ | ArN₂⁺ → ArX | | **Gattermann** | Cu/HCl | ArN₂⁺ → ArCl | | **Hofmann bromamide** ⭐ | Br₂/KOH | RCONH₂ → RNH₂ (−1 C!) | | **Gabriel** ⭐ | phthalimide/KOH | → 1° amine only | | **Carbylamine** | CHCl₃+KOH | 1° amine → foul isocyanide | | **Wurtz** | Na/dry ether | 2RX → R–R | | **Finkelstein** | NaI/acetone | RCl → RI | | **Swarts** | AgF | RCl → RF | | **HVZ** | X₂/Red P | Acid → α-halo acid | | **Friedel-Crafts** | RCl/AlCl₃ | Benzene → alkyl benzene | | **Coupling** | NaOH, 0–5°C | Diazonium + phenol/amine → azo dye | --- ## P. QUALITATIVE TESTS 🟢 | Test | Positive Result | |------|-----------------| | Tollen's | Silver mirror (aldehyde) | | Fehling's | Red ppt (aliphatic aldehyde) | | Lucas | 3° instant, 2° in 5 min, 1° no turbidity | | Hinsberg | Distinguishes 1°/2°/3° amines | | NaHCO₃ | Effervescence (COOH) | | 2,4-DNP | Orange/yellow ppt (C=O) | --- # 📐 PART 3: MATHEMATICS ## 1. Quadratic 🟡 S = −b/a, P = c/a | D = b²−4ac | Range extremum at x = −b/2a → (4ac−b²)/4a ## 2. Complex Numbers 🟡 | Concept | Formula | |---------|---------| | Euler ⭐ | e^(iθ) = cosθ + i sinθ | | Rotation ⭐ | z' = z₁ + (z₂−z₁)e^(iθ) | | Cube roots | 1+ω+ω² = 0, ω³ = 1 | | Identity | \|z₁+z₂\|² + \|z₁−z₂\|² = 2(\|z₁\|²+\|z₂\|²) | ## 3. Matrices & Determinants 🔴 | Concept | Formula | |---------|---------| | Inverse ⭐ | A⁻¹ = adjA/\|A\| | | adjoint ⭐ | \|adjA\| = \|A\|^(n−1); A·adjA = \|A\|I | | Skew-symmetric | Odd order ⇒ det = 0 | | Characteristic eq (2×2) ⭐ | A² − (trA)A + \|A\|I = O | | System ⭐ | \|A\|≠0 → unique; \|A\|=0 & (adjA)B≠0 → no solution; = 0 → infinite | ## 4. P&C + Binomial 🔴 | Concept | Formula | |---------|---------| | Circular ⭐ | (n−1)!; Necklace: (n−1)!/2 | | Derangements ⭐ | !4 = 9, !5 = 44 | | Selections w/ repetition | C(n+r−1, r) | | ΣnCr² | = (2n)!/(n!)² | | Binomial term ⭐ | T_{r+1} = ⁿCᵣ a^(n−r) bʳ | | Sum of coefficients | Put x = 1 (and x = −1 for alternating) | ## 5. Sequences & Series 🟡 | Series | Sum | |--------|-----| | Σn, Σn², Σn³ ⭐ | n(n+1)/2, n(n+1)(2n+1)/6, [n(n+1)/2]² | | Infinite GP | a/(1−r) | | AGP (infinite) ⭐ | a/(1−r) + dr/(1−r)² | | AM·HM | = GM² | ## 6. Trigonometry 🟡 ⭐ General solutions: sinθ = sinα → nπ + (−1)ⁿα; cosθ = cosα → 2nπ ± α; tanθ = tanα → nπ + α ⭐ a sinθ + b cosθ max/min = ±√(a²+b²) ⭐ **Triangle:** Δ = ½ab sinC = abc/4R = rs; R = a/2sinA; r = 4R sin(A/2)sin(B/2)sin(C/2) ## 7. Straight Lines & Circles 🔴 | Concept | Formula | |---------|---------| | Point-line distance ⭐ | \|ax₁+by₁+c\|/√(a²+b²) | | Circle general ⭐ | x²+y²+2gx+2fy+c = 0; r = √(g²+f²−c) | | Tangent slope form ⭐ | y = mx ± r√(1+m²) | | Orthogonal circles | 2g₁g₂+2f₁f₂ = c₁+c₂ | | Radical axis | S₁ − S₂ = 0 | ## 8. Conic Sections 🔴 | Conic | Focus | Eccentricity | Latus rectum | Tangent (slope m) | |-------|-------|--------------|--------------|-------------------| | y²=4ax | (a,0) | e=1 | 4a | y = mx + a/m | | x²/a²+y²/b²=1 | (±ae,0) | √(1−b²/a²) | 2b²/a | y = mx ± √(a²m²+b²) | | x²/a²−y²/b²=1 | (±ae,0) | √(1+b²/a²) | 2b²/a | y = mx ± √(am²−b²) | ⭐ Ellipse: PS + PS' = 2a | Hyperbola: PS − PS' = 2a | Rectangular hyperbola e = √2 ## 9. Limits & Continuity 🟡 ⭐ Standard limits: sinx/x → 1, (eˣ−1)/x → 1, ln(1+x)/x → 1, (1+x)^(1/x) → e ⭐ 1^∞ form: lim fᵍ = e^(lim g(f−1)) ## 10. Differentiation + AOD 🔴 | Concept | Formula | |---------|---------| | Parametric ⭐ | dy/dx = (dy/dt)/(dx/dt) | | Normal slope | −dx/dy | | Increasing ⭐ | f'(x) > 0 | | Maxima/minima ⭐ | f' = 0 + f'' < 0 (max), f'' > 0 (min) | | Absolute extrema | Check endpoints + critical points | ## 11. Integration + Area 🔴 ⭐ **Must-memorize:** - ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a) | ∫dx/√(a²−x²) = sin⁻¹(x/a) - ∫secx dx = ln\|secx+tanx\| | ∫eˣ(f+f') = eˣf + C - **King's property ⭐:** ∫₀ᵃf(x)dx = ∫₀ᵃf(a−x)dx - Odd function on [−a,a] → 0 - Area = ∫(y_upper − y_lower)dx ## 12. Differential Equations 🟡 ⭐ Linear: dy/dx + Py = Q → y·(IF) = ∫Q(IF)dx, IF = e^∫Pdx ## 13. Vectors & 3D 🟣 *(2-3 guaranteed Qs)* | Concept | Formula | |---------|---------| | Projection | a·b/\|b\| | | Area (triangle) ⭐ | ½\|a×b\| | | Volume ⭐ | Parallelepiped = \|[a b c]\|; Tetrahedron = (1/6)\|[abc]\| | | Vector triple product | a×(b×c) = (a·c)b − (a·b)c | | Point-plane distance ⭐ | \|ax₀+by₀+cz₀+d\|/√(a²+b²+c²) | | Line-plane angle | sinθ = \|b·n\|/(\|b\|\|n\|) | | Skew lines ⭐ | SD = \|(a₂−a₁)·(b₁×b₂)\|/\|b₁×b₂\| | | l²+m²+n² | = 1 | ## 14. Probability + Statistics 🔴 | Concept | Formula | |---------|---------| | Bayes ⭐ | P(Eᵢ/A) = P(Eᵢ)P(A/Eᵢ)/ΣP(Eⱼ)P(A/Eⱼ) | | Binomial ⭐ | P(X=r) = ⁿCᵣpʳq^(n−r); Mean = np, Var = npq | | At least one | 1 − (1−p)ⁿ | | Combined variance ⭐ | σ² = (n₁σ₁²+n₂σ₂²+n₁d₁²+n₂d₂²)/(n₁+n₂) | --- # 📊 PYQ WEIGHTAGE SUMMARY (2021–2025 Shifts) ## PHYSICS (out of ~25 Q) | Chapter | Avg Qs | Priority | |---------|--------|----------| | Modern Physics + Semiconductors | 3–4 | 🟣 | | Electrostatics + Capacitors | 2–3 | 🔴 | | Current Electricity | 2–3 | 🔴 | | Magnetism | 2 | 🔴 | | EMI + AC | 2 | 🟣 | | Ray Optics | 2 | 🔴 | | Thermodynamics + KTG | 2 | 🔴 | | Rotation | 1–2 | 🔴 | | SHM + Waves | 2 | 🟡 | | Kinematics+NLM+WPE+COM | 3–4 | 🟡 | | Wave Optics | 1 | 🟡 | | EM Waves + Units | 2 | 🟡 | ## CHEMISTRY | Chapter | Avg Qs | Priority | |---------|--------|----------| | Physical numericals (all) | 6–7 | 🟣 | | Organic (GOC + named rxns) | 5–6 | 🟣 | | Coordination | 2 | 🟣 | | Chemical Bonding | 2 | 🟣 | | p-block + d/f block | 2–3 | 🟡 | | Biomolecules | 1 | 🟡 | | Atomic structure + Periodicity | 2 | 🟡 | ## MATHS | Chapter | Avg Qs | Priority | |---------|--------|----------| | Calculus (entire) | 7–8 | 🟣 | | Coordinate Geometry | 3–4 | 🔴 | | Vectors + 3D | 2–3 | 🟣 | | Matrices + Determinants | 2 | 🔴 | | P&C + Binomial | 2 | 🔴 | | Probability + Statistics | 2 | 🔴 | | Algebra rest + Trigonometry | 3–4 | 🟡 | --- # 🔢 GOLDEN DATA TABLE (Memorize!) ## Physics Constants | Constant | Value | |----------|-------| | h | 6.63×10⁻³⁴ Js | | hc | **1240 eV·nm** ⚡ (saves time!) | | e | 1.6×10⁻¹⁹ C | | mₑ | 9.1×10⁻³¹ kg | | N_A | 6.022×10²³ | | 1 amu | **931.5 MeV** | | R (gas) | 8.314 J/mol·K = 0.0821 L·atm | | σ (Stefan) | 5.67×10⁻⁸ | | 1 atm | 1.013×10⁵ Pa | ## Maths Quick Data | Value | Amount | |-------|--------| | √2, √3, √5 | 1.414, 1.732, 2.236 | | ln2, ln10 | 0.693, 2.303 | | e | 2.718; 1/e ≈ 0.37 | | Factorials | 4! = 24, 5! = 120, 6! = 720, 7! = 5040 | | 2ⁿ | 512, 1024 (n=9,10) | ## Chemistry Quick Data | Item | Value | |------|-------| | F (Faraday) | 96500 C | | 0.059 | Nernst constant at 298 K | | Kw | 10⁻¹⁴ | | Flame colours | Na=Yellow, K=Violet, Ca=Brick red, Ba=Apple green, Cu=Blue-green | --- # ⚠️ SYLLABUS ALERT (2025 revision — verify NTA for 2027) **❌ DELETED:** Communication Systems, Transistors, X-rays (Physics) | States of Matter, Surface Chemistry, Hydrogen, s-Block, Metallurgy, Polymers, Chemistry in Everyday Life, Environmental (Chemistry) | Mathematical Induction, Mathematical Reasoning (Maths) --- # 💡 TOPPER STRATEGY (For 2027) 1. **📅 Revision cycle:** Formula sheet → D+1, D+3, D+7, D+21 (spaced repetition) 2. **📕 NCERT is non-negotiable** for Inorganic + Biomolecules (NTA lifts lines directly) 3. **📝 Solve last 5 years' PYQs shift-wise** — ~40% questions repeat concepts 4. **🎯 Attempt order:** Chemistry → Physics → Maths (bank marks early) 5. **📊 Target:** 99+ percentile ≈ 190–220 marks | Top NIT CSE ≈ 98.5+ percentile 6. **✍️ Error notebook:** Log every wrong question with the formula missed 7. **⏱️ Weekly:** 1 full mock + 2 subject-wise tests from Class 11 itself --- ## 🚀 WANT MORE? This is your **high-yield master sheet**. Reply with any of these and I'll build the deep-dive: - 📘 *"Expand Electrostatics"* / *"Expand Coordination Compounds"* — full chapter with every formula, trick & 20 PYQs - 🧪 *"All 100 organic reactions with mechanisms"* - 📐 *"Complete Calculus formula bank"* - 📊 *"Month-by-month study plan for 2027"* **Which chapter should I break down first?** 💪