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JEE Main · Formula sheets

JEE Main formula sheets by chapter

Each sheet is the "Formulas and facts to remember" list of Shishya's NCERT Class 11 or 12 notes for the chapter, word for word. Only notes whose fact-check was applied are used (checked 21 Oct 2026; an AI-assisted check against cited sources — the errors it found were corrected; some errors may remain). A chapter without a checked note is left out. The NCERT textbook is the authority.

Chapters are grouped under the syllabus unit they belong to on Shishya's syllabus page for this exam. JEE Main syllabus

Physics

Gravitation · NCERT Class 11

  • Newton's law of universal gravitation:F = G × m₁ × m₂ / r²G = 6.67 × 10⁻¹¹ N m² kg⁻² (universal gravitational constant)
  • Acceleration due to gravity at Earth's surface:g = G × M / R²where M is Earth's mass and R is Earth's radius; g ≈ 9.8 m/s²
  • Variation of g with altitude h (when h << R):g' = g × (1 − 2h/R)
  • Variation of g with depth d:g' = g × (1 − d/R)
  • Gravitational potential energy:U = −G × M × m / rNegative sign indicates bound system; zero at infinity.
  • Orbital velocity for circular orbit:v₀ = √(G × M / r) = √(g × R) for orbit near surface
  • Escape velocity from Earth's surface:vₑ = √(2 × G × M / R) = √(2 × g × R) ≈ 11.2 km/s
  • Kepler's third law:T² ∝ a³, or T² = (4π² / G × M) × a³where T is orbital period and a is semi-major axis.
  • Time period of satellite in circular orbit:T = 2π × √(r³ / G × M)

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Mechanical Properties of Solids · NCERT Class 11

  • Stress = Force / Area = F/A (unit: Pa or N/m²).
  • Longitudinal strain = Change in length / Original length = ΔL/L (no unit).
  • Volume strain = Change in volume / Original volume = ΔV/V (no unit).
  • Shear strain = Angular displacement θ (in radians) ≈ tan θ for small angles.
  • Young's modulus: Y = (F × L) / (A × ΔL).
  • Bulk modulus: B = −ΔP / (ΔV/V) = −P × V / ΔV.
  • Shear modulus: G = (F/A) / θ.
  • Elastic potential energy stored per unit volume = (1/2) × stress × strain = (1/2) × Y × (strain)².

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Thermal Properties of Matter · NCERT Class 11

  • Formula: T(K) = T(°C) + 273.15 · Meaning: Conversion from Celsius to Kelvin
  • Formula: T(°F) = (9/5)T(°C) + 32 · Meaning: Conversion from Celsius to Fahrenheit
  • Formula: ΔL = αL₀ΔT · Meaning: Linear expansion of a solid
  • Formula: ΔV = γV₀ΔT, γ ≈ 3α · Meaning: Volume expansion; γ is about three times α
  • Formula: Q = mcΔT · Meaning: Heat absorbed or released during temperature change
  • Formula: Q = mL · Meaning: Heat absorbed or released during phase change
  • Formula: H = kA(T₁ − T₂)/d · Meaning: Rate of heat conduction through a slab (Fourier's law)
  • Formula: P = σAT⁴ · Meaning: Power radiated by a black body (Stefan–Boltzmann law); σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
  • Water has an unusually high specific heat capacity (about 4186 J kg⁻¹ K⁻¹), which moderates climate near coasts.
  • Anomalous expansion of water: water is densest at 4 °C, so lakes freeze from the top down and aquatic life survives in winter.
  • Good conductors (metals) have high k; insulators (wood, air, wool) have low k.

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Thermodynamics · NCERT Class 11

  • First law: ΔU = Q − W (sign convention: Q positive when heat enters, W positive when system does work on surroundings).
  • Work done by an ideal gas at constant pressure: W = P ΔV.
  • Work in an isothermal expansion from V₁ to V₂: W = nRT ln(V₂ / V₁).
  • For an adiabatic process with an ideal gas: PV^γ = constant and TV^(γ−1) = constant, where γ = Cₚ / Cᵥ.
  • Relation between molar heat capacities: Cₚ − Cᵥ = R (for one mole of an ideal gas).
  • Carnot efficiency: η = 1 − T_cold / T_hot (temperatures in kelvin).
  • Coefficient of performance of a refrigerator: COP = Q_cold / W = T_cold / (T_hot − T_cold) for an ideal Carnot refrigerator.
  • Second law (Kelvin–Planck form): It is impossible to construct a heat engine that, operating in a cycle, converts all absorbed heat into work.

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Oscillations · NCERT Class 11

  • Expression: T = 1/f · Meaning: Time period (s) equals reciprocal of frequency (Hz).
  • Expression: ω = 2πf = 2π/T · Meaning: Angular frequency in rad/s.
  • Expression: x = A cos(ωt + φ) · Meaning: Displacement in SHM; A is amplitude, φ is phase constant.
  • Expression: v = dx/dt = −Aω sin(ωt + φ) · Meaning: Velocity in SHM; maximum value v_max = Aω.
  • Expression: a = dv/dt = −ω²x · Meaning: Acceleration in SHM; always opposite to displacement.
  • Expression: T (spring) = 2π√(m/k) · Meaning: Time period of a mass–spring oscillator.
  • Expression: T (pendulum) = 2π√(L/g) · Meaning: Time period of a simple pendulum (small angle).
  • Expression: E = (1/2)kA² = (1/2)mω²A² · Meaning: Total mechanical energy in SHM (constant).
  • Expression: KE = (1/2)mv² ; PE = (1/2)kx² · Meaning: Kinetic and potential energies; their sum equals E.

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Current Electricity · NCERT Class 12

  • Current: I = Q/t (charge per unit time).
  • Drift velocity relation: I = n A e vd.
  • Ohm's law: V = I R.
  • Resistance from resistivity: R = ρ L/A.
  • Temperature dependence of resistivity: ρ(T) = ρ₀ [1 + α (T − T₀)], where α is the temperature coefficient.
  • Series combination: Req = R₁ + R₂ + …
  • Parallel combination: 1/Req = 1/R₁ + 1/R₂ + …
  • Terminal voltage of a cell: V = ε − I r.
  • Wheatstone bridge balance condition: P/Q = R/S (no current through the galvanometer).
  • Potentiometer principle: potential drop along a uniform wire is proportional to length; used to compare emfs or measure internal resistance.

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Electromagnetic Waves · NCERT Class 12

  • Displacement current: Id = ε₀ × (dΦE/dt), where ΦE is electric flux. It has the same unit (ampere) as conduction current.
  • Speed of EM waves in vacuum: c = 1/√(μ₀ε₀) ≈ 3.00 × 10⁸ m/s.
  • Relation between field amplitudes: E₀ = c × B₀, or at any instant E = c × B.
  • Wave equation form: E = E₀ sin(kx − ωt), B = B₀ sin(kx − ωt), with k = 2π/λ and ω = 2πν.
  • Energy density: Total average energy density u = ε₀E² (since electric and magnetic contributions are equal in a travelling wave).
  • Intensity: I = (1/2) × c × ε₀ × E₀² (average power per unit area).
  • Radiation pressure: p = I/c (complete absorption) or p = 2I/c (perfect reflection).
  • Spectrum order by increasing frequency: Radio < Microwave < Infrared < Visible < Ultraviolet < X-ray < Gamma ray.

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Ray Optics and Optical Instruments · NCERT Class 12

  • Mirror formula: 1/v + 1/u = 1/f; magnification m = −v/u.
  • Refraction at a single spherical surface: n₂/v − n₁/u = (n₂ − n₁)/R.
  • Thin-lens formula: 1/v − 1/u = 1/f; magnification m = v/u.
  • Lens-maker's equation: 1/f = (n − 1)(1/R₁ − 1/R₂).
  • Power: P = 1/f (f in metres, P in dioptres). For lenses in contact, P_net = P₁ + P₂ + …
  • Critical angle: sin θ_c = 1/n when light goes from medium of index n to air.
  • Prism deviation: At minimum deviation D_m, the ray inside the prism is parallel to the base; n = sin[(A + D_m)/2] / sin(A/2), where A is the prism angle.
  • Simple microscope magnifying power: m = 1 + D/f (image at near point) or m = D/f (image at infinity), where D = 25 cm.
  • Compound microscope magnifying power: m ≈ (L/f_o)(D/f_e), with L the tube length.
  • Astronomical telescope magnifying power: m = f_o/f_e; tube length ≈ f_o + f_e for normal adjustment.

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Atoms · NCERT Class 12

  • Bohr radius: a₀ = ε₀ h² / (π m e²) ≈ 5.29 × 10⁻¹¹ m.
  • Radius of nth orbit: rₙ = n² a₀ / Z (Z = atomic number; for hydrogen Z = 1).
  • Velocity in nth orbit: vₙ = (Z e²) / (2 ε₀ n h) = (c α Z) / n, where α ≈ 1/137 is the fine-structure constant.
  • Total energy of nth level: Eₙ = −(m e⁴ Z²) / (8 ε₀² h² n²) = −13.6 Z² / n² eV.
  • Rydberg formula for wavelengths: 1/λ = R Z² (1/n₁² − 1/n₂²), n₂ > n₁.
  • Frequency of emitted photon: ν = (Eₙ₂ − Eₙ₁) / h.
  • Kinetic and potential energy in orbit: K = −E (positive), U = 2 E (negative), so total E = K + U = −K.
  • Distance of closest approach (α-particle): d = 2 k Z e² / (½ m v²), giving nuclear size upper limit.

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Semiconductor Electronics: Materials, Devices and Simple Circuits · NCERT Class 12

  • Conductivity of a semiconductor: σ = e (neμe + nhμh), where ne and nh are electron and hole concentrations, μe and μh are their mobilities, and e is the electronic charge.
  • Mass-action law (at thermal equilibrium): ne × nh = ni², where ni is the intrinsic carrier concentration.
  • Diode current (ideal): I = I0 [exp(eV / kT) − 1], where I0 is the reverse saturation current, V is the applied voltage, k is Boltzmann's constant and T is absolute temperature.
  • Knee voltage: Approximately 0.7 V for silicon diodes, 0.3 V for germanium diodes.
  • Current gain of a transistor (common-emitter): β = Ic / Ib, typically 20–500.
  • Relation between α and β: β = α / (1 − α), where α = Ic / Ie is the common-base current gain (slightly less than 1).
  • Zener regulation: The Zener diode maintains nearly constant output voltage Vz across a load even if input voltage or load current varies, provided it stays in the breakdown region.
  • Logic gate truth tables: NOT inverts (0→1, 1→0); AND gives 1 only when all inputs are 1; OR gives 1 when at least one input is 1; NAND and NOR are complements of AND and OR.

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Chemistry

Some Basic Concepts of Chemistry · NCERT Class 11

  • Number of moles: n = m / M, where m is mass in grams, M is molar mass in g mol⁻¹.
  • Number of particles: N = n × Nₐ, with Nₐ = 6.022 × 10²³ mol⁻¹.
  • Molar volume of an ideal gas at STP (0 °C, 1 bar): 22.7 L mol⁻¹ (or 22.4 L mol⁻¹ at older STP of 1 atm).
  • Percentage composition: (mass of element in 1 mol / molar mass of compound) × 100.
  • Molarity (M): moles of solute per litre of solution, unit mol L⁻¹.
  • Molality (m): moles of solute per kilogram of solvent, unit mol kg⁻¹.
  • Mass percent: (mass of solute / mass of solution) × 100.
  • Significant figures rule for multiplication/division: Result keeps as many significant figures as the quantity with the fewest.

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Structure of Atom · NCERT Class 11

  • Formula or fact: c = νλ · Meaning: Speed of light equals frequency times wavelength.
  • Formula or fact: E = hν = hc/λ · Meaning: Energy of a photon depends on its frequency.
  • Formula or fact: λ = h/mv · Meaning: de Broglie wavelength of a particle with mass m and velocity v.
  • Formula or fact: Δx × Δp ≥ h/4π · Meaning: Heisenberg uncertainty relation for position and momentum.
  • Formula or fact: Eₙ = −13.6/n² eV · Meaning: Energy of an electron in the nth orbit of hydrogen (Bohr model).
  • Formula or fact: rₙ = 0.529 × n² Å · Meaning: Radius of nth Bohr orbit for hydrogen.
  • Formula or fact: Aufbau principle · Meaning: Electrons fill orbitals in order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, …
  • Formula or fact: Pauli exclusion principle · Meaning: No two electrons in an atom can have the same set of all four quantum numbers.
  • Formula or fact: Hund's rule · Meaning: Electrons occupy degenerate orbitals singly first, with parallel spins, before pairing.

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Classification of Elements and Periodicity in Properties · NCERT Class 11

  • Effective nuclear charge (Z_eff): Z_eff = Z − σ, where Z is atomic number and σ is shielding constant. It measures the net positive charge felt by a valence electron.
  • Ionisation enthalpy (Δ_i H): Minimum energy required to remove the most loosely bound electron from an isolated gaseous atom in its ground state. Unit: kJ mol⁻¹.
  • Electron gain enthalpy (Δ_eg H): Enthalpy change when an electron is added to an isolated gaseous atom. A negative value means energy is released.
  • Electronegativity: A measure of an atom's ability to attract shared electrons in a covalent bond. Pauling scale is commonly used; fluorine = 4.0.
  • Covalent radius: Half the distance between nuclei of two identical atoms bonded covalently.
  • Periods and blocks: Period 1 has 2 elements (1s); periods 2 and 3 have 8 elements each (s and p blocks); period 4 has 18 elements (s, p and d blocks).
  • Diagonal relationship: Certain pairs (Li–Mg, Be–Al, B–Si) show similar properties because the increase in nuclear charge across a period offsets the increase in size down a group.

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Chemical Bonding and Molecular Structure · NCERT Class 11

  • Bond order (MOT) = (Nb − Na) / 2, where Nb = electrons in bonding MOs, Na = electrons in antibonding MOs. Higher bond order means stronger, shorter bond.
  • Formal charge = Valence electrons of free atom − [Non-bonding electrons + ½(Bonding electrons)].
  • Dipole moment μ = q × d, where q is charge magnitude and d is separation distance. SI unit is coulomb-metre; often given in debye (1 D ≈ 3.336 × 10⁻³⁰ C·m).
  • Hybridisation and geometry:
  • sp → linear, 180° (BeCl₂, CO₂)
  • sp² → trigonal planar, 120° (BF₃, C₂H₄)
  • sp³ → tetrahedral, 109.5° (CH₄, NH₄⁺)
  • sp³d → trigonal bipyramidal, 90° and 120° (PCl₅)
  • sp³d² → octahedral, 90° (SF₆)
  • VSEPR repulsion order: Lone pair–lone pair > lone pair–bond pair > bond pair–bond pair.
  • Sigma (σ) bond: Formed by head-on overlap of orbitals along the internuclear axis.
  • Pi (π) bond: Formed by sideways overlap of p orbitals above and below the internuclear axis.
  • Resonance: When a single Lewis structure cannot represent actual electron distribution, we draw multiple structures; the real molecule is a hybrid.

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Equilibrium · NCERT Class 11

  • Kc = [products]^coefficients / [reactants]^coefficients (each concentration in mol L⁻¹).
  • Kp = Kc (RT)^Δn, with Δn = sum of gaseous product coefficients minus sum of gaseous reactant coefficients.
  • pH = −log₁₀[H⁺]; pOH = −log₁₀[OH⁻]; pH + pOH = 14 (at 25 °C).
  • Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ at 25 °C.
  • For a weak acid HA with dissociation constant Ka, degree of dissociation α ≈ √(Ka / C) when α is small.
  • Henderson–Hasselbalch equation for a buffer: pH = pKa + log₁₀([salt]/[acid]).
  • Ksp for AgCl ⇌ Ag⁺ + Cl⁻ is Ksp = [Ag⁺][Cl⁻]; precipitation begins when ionic product > Ksp.

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Organic Chemistry – Some Basic Principles and Techniques · NCERT Class 11

  • General formula of alkanes: CₙH₂ₙ₊₂ (saturated hydrocarbons with only single bonds).
  • General formula of alkenes: CₙH₂ₙ (one carbon–carbon double bond).
  • General formula of alkynes: CₙH₂ₙ₋₂ (one carbon–carbon triple bond).
  • Degree of unsaturation (DBE): (2C + 2 + N − H − X) / 2, where C = carbons, N = nitrogens, H = hydrogens, X = halogens; counts rings plus multiple bonds.
  • Rf value in chromatography: Rf = (distance moved by substance) / (distance moved by solvent front); a characteristic ratio used to identify compounds.
  • Boiling-point trend: In a homologous series, boiling point rises by roughly 20–30 °C for each added –CH₂– unit because of increasing molecular mass and surface area.
  • Electronegativity order affecting polarity: F > O > N > C; bonds to more electronegative atoms are polar and influence solubility and reactivity.
  • Inductive effect: Electron-withdrawing groups (–NO₂, –Cl) pull electron density through sigma bonds; electron-donating groups (–CH₃) push electron density; both affect acidity and basicity.

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Hydrocarbons · NCERT Class 11

  • Alkane general formula: CₙH₂ₙ₊₂. Each successive member differs by CH₂ (homologous series).
  • Degree of unsaturation (DBE): For CₓHᵧ, DBE = (2x + 2 − y) / 2. One double bond or one ring equals one DBE; one triple bond equals two.
  • Markovnikov's rule: In the addition of HX to an unsymmetrical alkene, H goes to the carbon with more H atoms; the halide goes to the carbon with fewer H atoms.
  • Anti-Markovnikov (peroxide effect): With HBr in the presence of peroxides, the opposite orientation occurs via a free-radical mechanism. HCl and HI do not show this effect.
  • Benzene resonance energy: Benzene is about 150 kJ mol⁻¹ more stable than a hypothetical "cyclohexatriene" with localised double bonds. This extra stability explains its preference for substitution over addition.
  • Hückel's rule for aromaticity: A planar, cyclic, fully conjugated system is aromatic if it has (4n + 2) π electrons (n = 0, 1, 2, …).
  • Combustion of hydrocarbons: Complete combustion gives CO₂ and H₂O. Example: CH₄ + 2 O₂ → CO₂ + 2 H₂O.
  • Ozonolysis: Alkenes react with ozone followed by zinc/water to cleave the double bond and yield aldehydes or ketones; useful for locating the double bond position.

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The d-and f-Block Elements · NCERT Class 12

  • General electronic configuration of d-block: (n−1)d¹⁻¹⁰ ns¹⁻². For the first transition series (Sc to Zn), n = 4.
  • Exceptional configurations: Chromium is [Ar] 3d⁵ 4s¹ and copper is [Ar] 3d¹⁰ 4s¹ owing to the extra stability of half-filled and fully filled d-orbitals.
  • Oxidation state trend (first series): Maximum oxidation state increases from Sc (+3) to Mn (+7), then decreases towards Zn (+2). The +2 state arises from loss of two 4s electrons.
  • Standard electrode potential: E° values become less negative across the series, meaning later elements are weaker reducing agents. Copper has a positive E° for Cu²⁺/Cu, so it does not liberate H₂ from dilute acids.
  • Dichromate ion in acid: Cr₂O₇²⁻ + 14 H⁺ + 6 e⁻ → 2 Cr³⁺ + 7 H₂O. Orange dichromate is reduced to green Cr³⁺.
  • Permanganate ion in acid: MnO₄⁻ + 8 H⁺ + 5 e⁻ → Mn²⁺ + 4 H₂O. Purple permanganate becomes nearly colourless Mn²⁺.
  • Lanthanoid configuration: [Xe] 4f¹⁻¹⁴ 5d⁰⁻¹ 6s². The +3 state is most stable because removal of two 6s and one 4f or 5d electron gives a stable configuration.
  • Actinoid configuration: [Rn] 5f¹⁻¹⁴ 6d⁰⁻¹ 7s². Uranium commonly shows +6 (as in UO₂²⁺).

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Alcohols, Phenols and Ethers · NCERT Class 12

  • General formula of alcohols: CₙH₂ₙ₊₁OH (for saturated, mono-hydroxy alcohols).
  • Acidity order: alcohols (pKₐ ≈ 16–18) < water (pKₐ ≈ 15.7) < phenol (pKₐ ≈ 10).
  • Lucas test: tertiary alcohols react immediately with Lucas reagent (anhydrous ZnCl₂ + conc. HCl) giving a turbid layer; secondary alcohols react in 5–10 minutes; primary alcohols do not react at room temperature.
  • Iodoform test: ethanol and any secondary alcohol with a CH₃–CHOH– unit give a yellow precipitate of CHI₃ when warmed with I₂ and NaOH.
  • Kolbe reaction: phenol + NaOH → sodium phenoxide; then CO₂ at 400 K and 4–7 atm → sodium salicylate → salicylic acid on acidification.
  • Reimer–Tiemann reaction: phenol + CHCl₃ + aqueous NaOH → salicylaldehyde (ortho-hydroxybenzaldehyde).
  • Williamson synthesis: R–O⁻Na⁺ + R′–X → R–O–R′ + NaX (works best when R′–X is a primary halide to favour Sₙ2).
  • Dehydration temperature guide: ethanol requires about 443 K with excess conc. H₂SO₄ to form ethene; at 413 K diethyl ether forms instead.

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Amines · NCERT Class 12

  • General formula of amines: Primary amine R–NH₂, secondary amine R₂NH, tertiary amine R₃N.
  • IUPAC naming: Replace the ending -e of the parent alkane with -amine. Example: CH₃–NH₂ is methanamine; C₆H₅–NH₂ is aniline (common) or benzenamine (IUPAC).
  • Basicity order in gas phase: (CH₃)₃N > (CH₃)₂NH > CH₃NH₂ > NH₃ (inductive effect dominates).
  • Basicity in aqueous solution: (CH₃)₂NH > CH₃NH₂ > (CH₃)₃N > NH₃ (solvation effect also matters).
  • Reduction of nitro compounds: R–NO₂ + 3H₂ (Ni catalyst, or Sn/HCl, or Fe/HCl) → R–NH₂ + 2H₂O.
  • Gabriel synthesis: Potassium phthalimide + R–X → N-alkylphthalimide; then hydrolysis with aqueous NaOH or hydrazine gives R–NH₂.
  • Hoffmann bromamide degradation: R–CO–NH₂ + Br₂ + 4NaOH → R–NH₂ + Na₂CO₃ + 2NaBr + 2H₂O (amine has one carbon less).
  • Diazotisation: C₆H₅–NH₂ + NaNO₂ + 2HCl (0–5 °C) → C₆H₅–N₂⁺Cl⁻ + NaCl + 2H₂O.

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Biomolecules · NCERT Class 12

  • Glucose molecular formula: C₆H₁₂O₆ — an aldohexose with an aldehyde group at C-1.
  • Fructose molecular formula: C₆H₁₂O₆ — a ketohexose with a ketone group at C-2.
  • Peptide bond formation: amino acid 1 + amino acid 2 → dipeptide + H₂O (condensation reaction between –COOH of one and –NH₂ of another).
  • Isoelectric point: the pH at which an amino acid exists as a zwitterion with no net charge.
  • DNA double helix: two antiparallel strands held by hydrogen bonds; A pairs with T (two H-bonds), G pairs with C (three H-bonds).
  • Reducing sugars: carbohydrates with a free aldehyde or ketone group that can reduce Fehling's solution or Tollens' reagent (e.g., glucose, maltose).
  • Non-reducing sugars: sucrose is non-reducing because both anomeric carbons are involved in the glycosidic bond.
  • Vitamin deficiency examples: Vitamin A deficiency causes night blindness; Vitamin C deficiency causes scurvy; Vitamin D deficiency causes rickets.

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Mathematics

Relations and Functions · NCERT Class 11

  • Cartesian product: A × B = {(a, b) : a ∈ A and b ∈ B}. The number of elements is n(A) × n(B).
  • For any sets, A × B ≠ B × A in general (order matters in ordered pairs).
  • A × (B ∪ C) = (A × B) ∪ (A × C) and A × (B ∩ C) = (A × B) ∩ (A × C).
  • The number of relations from A to B is 2^(m×n) if A has m elements and B has n elements.
  • A function requires: every element of the domain has an image, and each element of the domain has only one image.
  • Domain of f(x) = 1/(x − 2) is all real numbers except x = 2.
  • Domain of f(x) = √(x − 3) is all real numbers x ≥ 3.
  • Range of f(x) = x² for x ∈ R is [0, ∞).

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Sequences and Series · NCERT Class 11

  • For an Arithmetic Progression with first term a and common difference d:
  • nth term: aₙ = a + (n − 1)d — each term is the first term plus (n − 1) times the common difference.
  • Sum of first n terms: Sₙ = n/2 × [2a + (n − 1)d] or equivalently Sₙ = n/2 × (a + l), where l is the last term.
  • Arithmetic mean of a and b: AM = (a + b)/2.
  • For a Geometric Progression with first term a and common ratio r:
  • nth term: aₙ = a × r^(n − 1) — each term is the first term multiplied by r raised to (n − 1).
  • Sum of first n terms: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1; Sₙ = na when r = 1.
  • Sum of infinite GP (|r| < 1): S∞ = a/(1 − r).
  • Geometric mean of two positive numbers a and b: GM = √(ab).
  • Relation between AM and GM:
  • For any two positive numbers a and b: AM ≥ GM, with equality only when a = b.

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Straight Lines · NCERT Class 11

  • Slope formula: m = (y₂ − y₁)/(x₂ − x₁)Meaning: Rise divided by run between two points.
  • Slope-intercept form: y = mx + cMeaning: m is the slope, c is the y-intercept (where the line crosses the y-axis).
  • Point-slope form: y − y₁ = m(x − x₁)Meaning: The line with slope m passing through the point (x₁, y₁).
  • Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)Meaning: The line passing through two given points (x₁, y₁) and (x₂, y₂).
  • Intercept form: x/a + y/b = 1Meaning: The line cutting the x-axis at (a, 0) and the y-axis at (0, b).
  • General form: Ax + By + C = 0Meaning: The most general way to write any straight line.
  • Angle between two lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|, provided m₁m₂ ≠ −1.Meaning: Gives the acute angle between two non-perpendicular lines.
  • Distance from point (x₁, y₁) to line Ax + By + C = 0: d = |Ax₁ + By₁ + C|/√(A² + B²)Meaning: The perpendicular (shortest) distance from the point to the line.

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Statistics · NCERT Class 11

  • Measure: Range · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): Maximum − Minimum
  • Measure: Mean deviation about mean · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): M.D.(x̄) = (1/n) × Σ |xᵢ − x̄|
  • Measure: Variance · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): σ² = (1/n) × Σ (xᵢ − x̄)²
  • Measure: Standard deviation · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): σ = √[(1/n) × Σ (xᵢ − x̄)²]
  • Short-cut formula for variance (ungrouped):σ² = (1/n) × Σ xᵢ² − (x̄)²
  • For grouped data with frequencies fᵢ (total N = Σ fᵢ):
  • Variance: σ² = (1/N) × Σ fᵢ (xᵢ − x̄)²
  • Alternate form: σ² = (1/N) × Σ fᵢ xᵢ² − (x̄)²
  • Coefficient of variation:CV = (σ / x̄) × 100
  • Meaning: CV expresses standard deviation as a percentage of the mean, allowing comparison when means or units differ.

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Inverse Trigonometric Functions · NCERT Class 12

  • Function: sin⁻¹ x · Domain: [−1, 1] · Principal-value range: [−π/2, π/2]
  • Function: cos⁻¹ x · Domain: [−1, 1] · Principal-value range: [0, π]
  • Function: tan⁻¹ x · Domain: (−∞, ∞) · Principal-value range: (−π/2, π/2)
  • Function: cot⁻¹ x · Domain: (−∞, ∞) · Principal-value range: (0, π)
  • Function: sec⁻¹ x · Domain: (−∞, −1] ∪ [1, ∞) · Principal-value range: [0, π], excluding π/2
  • Function: cosec⁻¹ x · Domain: (−∞, −1] ∪ [1, ∞) · Principal-value range: [−π/2, π/2], excluding 0
  • Complementary identities (for x in the respective domains):
  • sin⁻¹ x + cos⁻¹ x = π/2
  • tan⁻¹ x + cot⁻¹ x = π/2
  • sec⁻¹ x + cosec⁻¹ x = π/2
  • Negative-argument relations:
  • sin⁻¹(−x) = −sin⁻¹ x
  • cos⁻¹(−x) = π − cos⁻¹ x
  • tan⁻¹(−x) = −tan⁻¹ x
  • Sum formula for tan⁻¹ (when xy < 1):
  • tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x + y)/(1 − xy)]
  • Double-angle form:
  • 2 tan⁻¹ x = sin⁻¹[2x/(1 + x²)] = cos⁻¹[(1 − x²)/(1 + x²)] = tan⁻¹[2x/(1 − x²)], valid for |x| ≤ 1 (sin⁻¹ form), x ≥ 0 (cos⁻¹ form) and −1 < x < 1 (tan⁻¹ form).

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Determinants · NCERT Class 12

  • det([[a, b], [c, d]]) = ad − bc — the product of the main diagonal minus the product of the other diagonal.
  • For a 3 × 3 matrix [[a₁, b₁, c₁], [a₂, b₂, c₂], [a₃, b₃, c₃]], expanding along the first row: |A| = a₁(b₂c₃ − b₃c₂) − b₁(a₂c₃ − a₃c₂) + c₁(a₂b₃ − a₃b₂).
  • |kA| = k^n × |A| for an n × n matrix A.
  • |AB| = |A| × |B| for square matrices of the same order.
  • |A^T| = |A| — the determinant of the transpose equals the determinant itself.
  • adj(A) × A = A × adj(A) = |A| × I, where I is the identity matrix.
  • A⁻¹ = (1/|A|) × adj(A), provided |A| ≠ 0.
  • Cramer's rule for two variables: if ax + by = e and cx + dy = f, then x = (ed − bf)/(ad − bc), y = (af − ec)/(ad − bc), when ad − bc ≠ 0.

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Continuity and Differentiability · NCERT Class 12

  • Formula / Rule: d/dx (xⁿ) = n x^(n−1) · Meaning: Power rule, valid for any real n.
  • Formula / Rule: d/dx (eˣ) = eˣ · Meaning: Exponential function is its own derivative.
  • Formula / Rule: d/dx (aˣ) = aˣ ln a · Meaning: Base-a exponential, a > 0, a ≠ 1.
  • Formula / Rule: d/dx (ln x) = 1/x · Meaning: Natural logarithm, x > 0.
  • Formula / Rule: d/dx (sin x) = cos x ; d/dx (cos x) = −sin x · Meaning: Basic trigonometric derivatives.
  • Formula / Rule: d/dx (tan x) = sec² x ; d/dx (cot x) = −cosec² x · Meaning:
  • Formula / Rule: d/dx (sin⁻¹ x) = 1/√(1 − x²) ; d/dx (cos⁻¹ x) = −1/√(1 − x²) · Meaning: Inverse sine and cosine, |x| < 1.
  • Formula / Rule: d/dx (tan⁻¹ x) = 1/(1 + x²) · Meaning: Inverse tangent, all real x.
  • Formula / Rule: Product rule: d/dx (uv) = u (dv/dx) + v (du/dx) · Meaning:
  • Formula / Rule: Quotient rule: d/dx (u/v) = [v (du/dx) − u (dv/dx)] / v² · Meaning:
  • Formula / Rule: Second derivative: d²y/dx² = d/dx (dy/dx) · Meaning: Measures rate of change of the slope.

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Vector Algebra · NCERT Class 12

  • Formula / Rule: |a| = √(a₁² + a₂² + a₃²) · Meaning: Magnitude of a vector.
  • Formula / Rule: Unit vector â = a / |a| · Meaning: A vector of magnitude 1 in the direction of a.
  • Formula / Rule: a · b = a₁b₁ + a₂b₂ + a₃b₃ · Meaning: Dot product via components.
  • Formula / Rule: a · b = |a| |b| cos θ · Meaning: Dot product via magnitudes and angle.
  • Formula / Rule: a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k · Meaning: Cross product in components.
  • Formula / Rule: |a × b| = |a| |b| sin θ · Meaning: Magnitude of cross product.
  • Formula / Rule: a · a = |a|² · Meaning: Useful identity for magnitude.
  • Formula / Rule: a × a = 0 · Meaning: Cross product of a vector with itself is the zero vector.
  • Formula / Rule: a · b = 0 ⇔ a ⊥ b (if both non-zero) · Meaning: Perpendicularity test.
  • Formula / Rule: a × b = 0 ⇔ a ∥ b (if both non-zero) · Meaning: Parallelism test.
  • Formula / Rule: [a b c] = a · (b × c) · Meaning: Scalar triple product (gives volume).

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From shishya.in — JEE Main formula sheets. Taken word for word from Shishya's fact-checked NCERT notes (written with AI); the NCERT textbook is the authority.