Physics
XC = 1/(ωC) and XL = ωL — Capacitor *cuts* frequency (inverse), inductor *loves* frequency (direct). At high ω: XC→0 (short), XL→∞ (open). Reversed at low ω.
RLC Resonance: ω₀ = 1/√(LC) — At resonance XL = XC, impedance Z = R (minimum), current maximum, power factor = 1. Circuit is purely resistive; voltage leads/lags vanish.
Radioactive decay N = N₀e^(-λt) — Half-life T½ = 0.693/λ. For each T½ elapsed, quantity halves: 1 → ½ → ¼ → ⅛. "HALF-ing the LIFE repeatedly."
Capacitors in series/parallel flip from resistors — Series: 1/C = 1/C₁ + 1/C₂ (reciprocals add). Parallel: C = C₁ + C₂ (direct add). Opposite of resistors.
ε₀μ₀c² = 1 — Speed of light c = 1/√(ε₀μ₀). Remember "Epsilon-Mu-C-squared equals One" to relate permittivity, permeability, and light speed in vacuum.
Lenz's Law: "Nature Opposes Change" — Induced current's magnetic field opposes the flux change causing it. Right-hand thumb rule: curl fingers in induced-current direction, thumb shows field.
Sky is Blue, Sunset is Red — Rayleigh scattering ∝ 1/λ⁴. Blue (short λ) scatters most (daytime sky). At sunset, blue scattered away, red (long λ) penetrates—reaches eyes.
Chemistry
Lucas Test: 1°-2°-3° alcohol reactivity — 3° reacts immediately (turbidity at once), 2° in ~5 min, 1° needs heating. "Three-Two-One, Faster to Done."
Aldehyde vs Ketone: Tollen's & Fehling's — Aldehydes reduce both (silver mirror, red ppt); ketones don't. "Alde-HYPE does the test, Ketone Takes REST."
Amine basicity order (aliphatic) — 2° > 1° > 3° in gas phase. In water: 2° > 1° > 3° due to steric hindrance and solvation. Remember "Two wins in Water."
Aufbau: n+l rule — Lower n+l fills first; if equal n+l, lower n first. Example: 4s (4+0=4) fills before 3d (3+2=5). "N-plus-L, then N."
Mole concept: 1 mole = 6.022×10²³ — Avogadro's number. Mass in grams = Molar mass for 1 mole. "602214076…billion billion" particles.
Markovnikov's Rule — In HX addition to alkene, H goes to carbon with more H already (richer get richer). Anti-Markovnikov with peroxide (ROOR) for HBr only.
SN1 vs SN2 — SN1: 3° > 2° > 1° (carbocation stability), racemization, protic solvent. SN2: 1° > 2° > 3° (steric access), inversion, aprotic solvent. "One is Opposite of Two."
Mathematics
Maxima/Minima: f'(x)=0, check f''(x) — If f''<0, local maximum (concave down). If f''>0, local minimum (concave up). "Negative is peak, Positive is pit."
Area under curve: ∫ y dx — Between curves: ∫(yupper - ylower)dx over [a,b]. Always subtract lower from upper to keep area positive.
— If n even: middle term is (n/2 + 1)th. If n odd: two middle terms are (n+1)/2 th and (n+3)/2 th.