JEE Advanced flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rolling without slipping condition

The contact point has zero instantaneous velocity, so center-of-mass speed equals angular speed times radius: v = ωR. This links translational and rotational kinetic energy.

Carnot efficiency formula

For a reversible engine between hot reservoir temperature T_H and cold reservoir temperature T_C, efficiency equals 1 − T_C/T_H, with both temperatures measured in kelvin. No real engine exceeds this.

Photoelectric stopping potential

Maximum kinetic energy of emitted electrons is K_max = hf − φ, where φ is the work function. Stopping potential V_0 satisfies eV_0 = K_max and is independent of light intensity.

Capacitance of parallel plate

For vacuum-separated plates of area A and separation d, C = ε_0 A/d. Inserting a dielectric of constant K increases capacitance to Kε_0 A/d and reduces the field inside.

Young double slit fringe width

Fringe width on a screen at distance D from slits separated by d is β = λD/d. Bright fringes occur at path difference nλ and dark fringes at (2n−1)λ/2.

Bohr radius formula

For a hydrogen-like atom of atomic number Z, the nth Bohr radius is r_n = 0.529 n²/Z angstroms. Energy is E_n = −13.6 Z²/n² electron volts.

Escape velocity from Earth

Minimum launch speed ignoring air resistance is v_e = √(2GM/R) = √(2gR), about 11.2 km/s at Earth's surface. It is independent of projectile mass.

Terminal velocity condition

A falling body reaches terminal velocity when drag plus buoyancy balances weight, giving zero acceleration. For Stokes drag, v_t = 2r²(ρ − σ)g/(9η).

Magnetic force on moving charge

A charge q moving with velocity v in magnetic field B experiences F = q(v × B). The force is perpendicular to both v and B, so magnetic force does no work.

Energy stored in inductor

An inductor carrying current I stores magnetic energy U = ½LI². During current change, induced emf is −L dI/dt, opposing the change according to Lenz's law.

Beat frequency

When two sound waves of nearby frequencies f1 and f2 superpose, intensity varies periodically at beat frequency |f1 − f2|. The perceived pitch is approximately the average frequency.

Steric number hybridization rule

Steric number equals sigma bonds plus lone pairs on the central atom. It predicts hybridization: 2 is sp, 3 is sp², 4 is sp³, 5 is sp³d, and 6 is sp³d².

Gibbs free energy spontaneity

At constant temperature and pressure, a process is spontaneous when ΔG = ΔH − TΔS is negative. Equilibrium corresponds to ΔG = 0, and ΔG° = −RT ln K.

Kp and Kc relation

For a gas-phase reaction, K_p = K_c(RT)^Δn, where Δn is moles of gaseous products minus gaseous reactants. R is in appropriate units and T is kelvin.

First order half-life

For a first-order reaction, half-life t_1/2 = ln2/k = 0.693/k, independent of initial concentration. After n half-lives, the remaining fraction is (1/2)^n.

Nernst equation at 298 K

Cell potential is E = E° − (0.0591/n) log Q at 298 K, where n is electrons transferred and Q is the reaction quotient. At equilibrium E becomes zero.

Octahedral CFSE formula

CFSE = (−0.4 x + 0.6 y)Δ_o, where x is the number of t2g electrons and y is the number of eg electrons. Paired electrons contribute additional pairing energy.

Markovnikov addition rule

In addition of HX to an unsymmetrical alkene, hydrogen adds to the carbon bearing more hydrogens, giving the more stable carbocation intermediate. Peroxides reverse HBr addition only.

Hund's rule of maximum multiplicity

Within a degenerate subshell, electrons fill orbitals singly with parallel spins before pairing. This minimizes electron-electron repulsion and maximizes total spin, giving the lowest energy term.

Henderson-Hasselbalch equation

For a buffer, pH = pK_a + log([conjugate base]/[weak acid]). It estimates pH when concentrations are moderate and the acid-base equilibrium dominates over water autoionization.

Degree of unsaturation formula

For a hydrocarbon with formula C_c H_h, degree of unsaturation = (2c + 2 − h)/2. Each ring or pi bond counts as one; halogens count as hydrogen and oxygen is ignored.

Arrhenius equation

Rate constant k = A e^(−E_a/RT), where A is the pre-exponential factor and E_a is activation energy. A plot of ln k versus 1/T is linear with slope −E_a/R.

Eccentricity of ellipse

For ellipse x²/a² + y²/b² = 1 with a > b, eccentricity e = √(1 − b²/a²). Foci lie at (±ae, 0) and directrices are x = ±a/e.

Conic discriminant test

For ax² + 2hxy + by² + 2gx + 2fy + c = 0, the conic is an ellipse if h² − ab < 0, a parabola if equal to zero, and a hyperbola if positive, when nondegenerate.

L'Hôpital indeterminate forms

L'Hôpital's rule applies to limits of the form 0/0 or ∞/∞. Differentiate numerator and denominator separately, then take the limit again if the result remains indeterminate.

Bayes theorem formula

P(A|B) = P(B|A)P(A)/P(B). For a partition A_i, the denominator is P(B) = Σ P(B|A_i)P(A_i), giving posterior probabilities from prior and likelihood.

Cube roots of unity

The solutions of z³ = 1 are 1, ω = e^(2πi/3), and ω². They satisfy 1 + ω + ω² = 0 and ω³ = 1, which simplifies many symmetric complex expressions.

Scalar triple product volume

The absolute value |a·(b×c)| equals the volume of the parallelepiped with edges a, b, c. Three vectors are coplanar exactly when this scalar triple product is zero.

Determinant of adjugate

For an n×n matrix A, det(adj A) = (det A)^(n−1). Also A(adj A) = (det A)I, so the inverse is adj A divided by det A when nonzero.

General term binomial expansion

In (a + b)^n, the term containing b^r is T_{r+1} = C(n,r) a^(n−r) b^r. Setting exponent conditions locates a specific term without full expansion.

Integration by parts formula

∫u dv = uv − ∫v du, choosing u by the ILATE priority order. Repeated application handles products of polynomials with exponentials, logarithms, or trigonometric functions.

AM-GM inequality

For nonnegative real numbers, arithmetic mean is at least geometric mean, with equality only when all numbers are equal. For two positives, a + b ≥ 2√(ab).