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JEE Main High-Yield Formulas and Physical Constants

Essential formulas, physical constants, shortcut algorithms, and reaction conditions tested across Physics, Chemistry, and Mathematics in JEE Main.

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Moment of Inertia of Standard Rigid Bodies
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1
Moment of Inertia of Standard Rigid Bodies
Ring: MR^2; Solid Disc: 1/2 MR^2; Solid Sphere: 2/5 MR^2; Spherical Shell: 2/3 MR^2; Rod about center: 1/12 ML^2; Rod about end: 1/3 ML^2.
2
Escape Velocity from Earth's Surface
v_e = sqrt(2 G M / R) = sqrt(2 g R) approx 11.2 km/s, independent of the mass or launch angle of the projected body.
3
Bernoulli's Equation for Incompressible Fluid Flow
P + 1/2 rho v^2 + rho g h = constant; expresses conservation of mechanical energy per unit volume along a streamline.
4
Carnot Engine Thermal Efficiency
eta = 1 - T_cold / T_hot = (Q_in - Q_out) / Q_in, where temperatures T must be evaluated strictly in absolute Kelvin scale.
5
Simple Harmonic Motion Velocity and Acceleration as Functions of Position
v(x) = +/- omega sqrt(A^2 - x^2); a(x) = -omega^2 x; maximum velocity occurs at mean position (x = 0), maximum acceleration at extremes (x = +/- A).
6
Doppler Effect for Sound Waves
f' = f * [(v +/- v_o) / (v -/+ v_s)], where upper signs correspond to motion towards each other (frequency increase) and lower signs to motion away.
7
Poynting Vector and Electromagnetic Wave Intensity
S = (1 / mu_0) (E x B); Intensity I = S_avg = 1/2 epsilon_0 c E_0^2 = (E_0 B_0) / (2 mu_0), directed along wave propagation vector k.
8
de Broglie Wavelength of an Electron Accelerated through Potential V
lambda = h / sqrt(2 m e * V) approx 1.227 / sqrt(V) nanometers (or 12.27 / sqrt(V) Angstroms).
9
Radioactive Decay Law and Activity
N(t) = N_0 e^(-lambda t); Activity A(t) = lambda N(t) = A_0 e^(-lambda * t); Half-life t_1/2 = ln(2) / lambda approx 0.693 / lambda.
10
Crystal Field Stabilization Energy (CFSE) for Octahedral Complexes
CFSE = [-0.4 n(t2g) + 0.6 n(eg)] Delta_o + m P, where P is pairing energy; strong-field ligands (CN-, CO) induce low-spin pairing (Delta_o > P).
11
Jahn-Teller Distortion Occurrence
Occurs in non-linear coordination complexes possessing orbitally degenerate electronic ground states, most prominently in high-spin d4 (Cr2+), low-spin d7 (Co2+), and d9 (Cu2+) octahedral geometries.
12
van 't Hoff Factor (i) for Association and Dissociation
Dissociation: i = 1 + (n - 1)alpha; Association: i = 1 + (1/n - 1)alpha, where alpha is the degree of dissociation/association.
13
Clapeyron-Clausius Equation for Vapor Pressure
ln(P2 / P1) = (Delta H_vap / R) * [1/T1 - 1/T2], describing exponential increase of equilibrium vapor pressure with absolute temperature.
14
Reimer-Tiemann Reaction Mechanism
Phenol + CHCl3 + aq. NaOH -> Salicylaldehyde (o-hydroxybenzaldehyde) via electrophilic attack of neutral dichlorocarbene (:CCl2) intermediate.
15
Kolbe's Carboxylation Reaction
Sodium phenoxide + CO2 (at 125 deg C, 4-7 atm) followed by acidification -> Salicylic acid (o-hydroxybenzoic acid), the key precursor for aspirin.
16
Rosenmund Reduction Reaction
Acyl chloride + H2 in the presence of Pd/BaSO4 poisoned with quinoline/sulfur -> Aldehyde; prevents over-reduction to primary alcohol.
17
Leibnitz Rule for Differentiation under Integral Sign
d/dx [integral from u(x) to v(x) of f(t) dt] = f(v(x)) v'(x) - f(u(x)) u'(x).
18
Distance Between Two Parallel Planes in 3D Space
For planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0: Distance d = |D1 - D2| / sqrt(A^2 + B^2 + C^2).
19
Standard Locus of Orthocenter and Circumcenter of Right Triangle
In any right-angled triangle: Orthocenter lies exactly at the vertex with the right angle; Circumcenter lies at the midpoint of the hypotenuse.
20
Eccentricity Formula for Ellipse and Hyperbola
Ellipse: e = sqrt(1 - b^2/a^2) < 1; Hyperbola: e = sqrt(1 + b^2/a^2) > 1, where a is semi-major (or transverse) and b is semi-minor (or conjugate) axis.
21
Condition for Common Roots in Two Quadratic Equations
One common root: (c1a2 - c2a1)^2 = (b1c2 - b2c1) (a1b2 - a2*b1); Both roots common: a1/a2 = b1/b2 = c1/c2.
22
Sum of Powers of Natural Numbers Formulas
Sum(n) = n(n+1)/2; Sum(n^2) = n(n+1)(2n+1)/6; Sum(n^3) = [n(n+1)/2]^2 = [Sum(n)]^2.
23
De Moivre's Theorem for Roots of Unity
The n nth roots of unity are given by e^(i 2pi k / n) for k = 0, 1, ..., n-1; their sum is strictly 0 and product is (-1)^(n-1).
24
Poisson Distribution Formula and Variance
P(X = k) = (lambda^k * e^(-lambda)) / k!; Mean = lambda, and Variance = lambda (mean equals variance in a Poisson process).
25
Fundamental Thermodynamic Identity for Free Energy
dG = VdP - SdT; at constant temperature (dT = 0), (dG/dP)_T = V; at constant pressure (dP = 0), (dG/dT)_P = -S.

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