JEE Advanced cheat sheet

Rolling without slipping: v_cm = ωR and a_cm = αR; total kinetic energy = ½mv_cm²(1 + I_cm/mR²). Static friction at the contact point does no work.

Projectile on level ground: range R = v² sin2θ/g, maximum height H = v² sin²θ/(2g), time of flight T = 2v sinθ/g. Complementary angles give equal range.

SHM: acceleration a = −ω²x; spring period T = 2π√(m/k); simple pendulum period T = 2π√(l/g); total energy E = ½mω²A².

Thin lens with Cartesian sign convention: 1/v − 1/u = 1/f; spherical mirror: 1/v + 1/u = 1/f. Magnification is m = v/u for a lens and m = −v/u for a mirror.

Young double slit: fringe width β = λD/d. Bright fringes occur at path difference nλ; dark fringes at (2n−1)λ/2. A medium of refractive index μ reduces β by factor μ.

Electrostatics: dipole field on axis is 2kp/r³ and equatorial is kp/r³; capacitor energy U = ½CV²; a dielectric slab of constant K multiplies capacitance by K.

Bohr model: E_n = −13.6 Z²/n² eV and r_n = 0.529 n²/Z Å. Angular momentum is quantized as L = nh/2π; photon energy equals the level difference.

Thermodynamics: first law ΔU = Q − W; Carnot efficiency η = 1 − T_C/T_H; for an ideal gas C_P − C_V = R and γ = C_P/C_V.

VSEPR: steric number = sigma bonds + lone pairs; 2 linear, 3 trigonal planar, 4 tetrahedral, 5 trigonal bipyramidal, 6 octahedral. In TBP, lone pairs occupy equatorial positions first.

Hybridization from steric number: 2 → sp, 3 → sp², 4 → sp³, 5 → sp³d, 6 → sp³d². Count a multiple bond as one electron domain.

Thermodynamics: ΔG = ΔH − TΔS; spontaneous at constant T and P when ΔG < 0. Also ΔG° = −RT ln K and ΔG° = −nFE°cell.

Equilibrium: K_p = K_c(RT)^Δn_gas, where Δn_gas is gaseous product moles minus gaseous reactant moles. Compare reaction quotient Q with K to predict direction; catalysts do not change K.

Kinetics: first-order t_1/2 = 0.693/k; Arrhenius k = A e^(−E_a/RT); a plot of ln k versus 1/T has slope −E_a/R.

Electrochemistry: Nernst equation E = E° − (0.0591/n) log Q at 298 K. Molar conductivity Λ_m = κ/c increases on dilution for weak electrolytes.

Coordination: octahedral CFSE = (−0.4 x + 0.6 y)Δ_o, where x is t2g electrons and y is eg electrons. Strong-field ligands favor low-spin configurations.

Organic addition: Markovnikov rule places H on the alkene carbon with more hydrogens; peroxide effect gives anti-Markovnikov HBr only. SN1 favors tertiary substrates and polar protic solvents; SN2 favors primary substrates and polar aprotic solvents.

Electrophilic aromatic substitution: −NH2, −OH, −OR, and alkyl groups are ortho/para directors; −NO2, −CN, −COOH, and −SO3H are meta directors; halogens are deactivating ortho/para directors.

Conic identification for ax² + 2hxy + by² + 2gx + 2fy + c = 0: ellipse if h² − ab < 0, parabola if h² − ab = 0, hyperbola if h² − ab > 0, for nondegenerate cases.

Standard conics: ellipse x²/a² + y²/b² = 1 has e = √(1 − b²/a²); hyperbola x²/a² − y²/b² = 1 has e = √(1 + b²/a²) and asymptotes y = ±(b/a)x.

L'Hôpital's rule applies only to 0/0 or ∞/∞ indeterminate forms; differentiate numerator and denominator separately and recheck. Rationalization or series expansion is often faster.

Definite integrals: ∫_a^b f(x)dx = ∫a^b f(a+b−x)dx; ∫−a^a f(x)dx = 0 for odd f and 2∫_0^a f(x)dx for even f.

Probability: P(A|B) = P(B|A)P(A)/P(B); total probability P(B) = Σ P(B|A_i)P(A_i) for a partition. Independent events satisfy P(A∩B) = P(A)P(B).

Complex numbers: |z1z2| = |z1||z2| and arg(z1z2) = arg z1 + arg z2. Cube roots of unity are 1, ω, ω² with 1 + ω + ω² = 0 and ω³ = 1.

Vectors: scalar triple product [a b c] = a·(b×c) equals signed parallelepiped volume; coplanar iff [a b c] = 0. Vector triple product: a×(b×c) = (a·c)b − (a·b)c.

Matrices: det(AB) = det A det B; det(adj A) = (det A)^(n−1) for n×n; A^(−1) = adj A/det A when det A ≠ 0. Eigenvalues solve det(A − λI) = 0.

Binomial: in (a+b)^n, T_{r+1} = C(n,r)a^(n−r)b^r. A middle term exists only for even n; the greatest coefficient is found by comparing consecutive term ratios.