Physics
XL < XC frequency rule — In AC circuits, inductive reactance XL increases with frequency (XL = 2πfL), capacitive reactance XC decreases (XC = 1/2πfC). Remember: "L goes up, C comes down" with frequency.
CIVIL for phase difference — In pure capacitor, Current leads Voltage (C-I-V). In pure inductor, Voltage leads Current (V-I-L). Helps remember phase relationships in AC circuits instantly.
Photoelectric work function — Maximum KE = hf - φ. Remember "High Frequency Minus Phi" — incoming photon energy minus work function equals electron's escape energy. Threshold means KE = 0.
Right-Hand Thumb Rule (Fleming's) — For electromagnetic induction: Thumb = motion, Forefinger = field, Middle finger = induced current. "TFM"順序 matches physical setup in generator problems.
Bohr radius formula — r = n²(0.529 Å)/Z. First orbit of hydrogen = 0.529 Å. Radius grows as square of shell number, shrinks with atomic number Z.
Nuclear binding energy — BE = Δm × 931.5 MeV. Mass defect in amu times 931.5 gives binding energy. Iron-56 has maximum BE per nucleon (~8.8 MeV), most stable.
Capacitor series/parallel — Series: add reciprocals (like resistors in parallel). Parallel: direct addition (like resistors in series). "Capacitors are opposite to resistors" for combination rules.
Chemistry
Lucas Test timing — Tertiary alcohols react immediately (cloudiness), secondary in 5 minutes, primary don't react at room temperature. Remember: "3°-instant, 2°-five, 1°-negative" for ZnCl₂/HCl test.
Aldehyde vs Ketone tests — Aldehydes give positive Tollen's (silver mirror) and Fehling's (red precipitate). Ketones don't. Mnemonic: "AldehydeS give Silver" — both start with S sound.
Basicity of amines — In gaseous phase: 3° > 2° > 1° > NH₃. In aqueous phase: 2° > 1° > 3° > NH₃. Remember solvation reverses the order for tertiary.
Aufbau principle order — 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s. Trick: n+l rule; lower n+l fills first, if equal then lower n first.
VSEPR shapes — 2 bp = linear (180°), 3 bp = trigonal planar (120°), 4 bp = tetrahedral (109.5°), 5 bp = trigonal bipyramidal, 6 bp = octahedral (90°).
Carbocation stability — 3° > 2° > 1° > CH₃⁺. Hyperconjugation and inductive effect: more alkyl groups = more stable. "Three is best, methyl is worst" for carbocation intermediates.
Chelate effect — Polydentate ligands (EDTA, en, ox) form more stable complexes than monodentate due to entropy gain. EDTA is hexadentate, forms very stable 1:1 complexes.
Mathematics
L'Hospital applicability — Only for 0/0 or ∞/∞ forms. Differentiate numerator and denominator separately (NOT quotient rule). Verify indeterminate form before applying.
Integration by parts — ∫u dv = uv - ∫v du. Choose u by ILATE: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. First in ILATE becomes u.
Binomial middle term — For (a+b)ⁿ, if n is even, middle term is (n/2 + 1)th term. If n is odd, two middle terms are (n+1)/2 and (n+3)/2. Greatest coefficient here.
Complex number modulus-argument — |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂). For division: moduli divide, arguments subtract. Remember "multiply adds, divide subtracts" for arguments.
Conic eccentricity — Circle e=0, Ellipse 0<e<1, Parabola e=1, Hyperbola e>1. Remember "CEPH" in increasing eccentricity order: Circle-Ellipse-Parabola-Hyperbola.
Differential equation order/degree — Order = highest derivative present. Degree = power of highest order derivative after removing radicals/fractions. Order always defined, degree may not exist.
Determinant row operations — Swapping rows changes sign. Multiplying row by k multiplies determinant by k. Adding multiple of one row to another keeps determinant unchanged.
Continuity conditions — Function continuous at x=a if: (1) f(a) exists, (2) lim(x→a) f(x) exists, (3) both are equal. All three conditions mandatory; missing one means discontinuous.