Physics
AC Phase Rule (XL–XC) — In series LCR: voltage leads current when XL > XC (inductive), lags when XC > XL (capacitive). Remember "CIVIL": in C (capacitor) I leads V, in L (inductor) V leads I.
Transformer Turns Ratio — Np/Ns = Vp/Vs = Is/Ip. Trick: "Voltage follows Turns, Current inverses" – step-up doubles voltage, halves current (power stays constant in ideal transformer).
Bohr Radius Formula — rₙ = 0.529n²/Z Ångstroms. Remember "0.529" as "May 29" (5/29) for ground-state hydrogen radius; scales with n² and inversely with atomic number Z.
Balmer Series Wavelength — Visible hydrogen lines: 656nm (red), 486nm (cyan), 434nm (blue), 410nm (violet). Mnemonic: "6-5, 4-8, 4-3, 4-1" – first digit descends, gives visible spectrum order.
Binding Energy per Nucleon Peak — Maximum at Fe-56 (Iron, mass number 56). Remember "Iron is the most stable" – fusion releases energy below Fe, fission above Fe.
Radioactive Decay N=N₀e⁻ᵏᵗ — After n half-lives, N = N₀/(2ⁿ). Quick check: 10 half-lives ≈ 1000-fold reduction (2¹⁰=1024), useful for carbon-dating problems.
Metre Bridge Balance Condition — (R₁/R₂) = (l₁/l₂) where l₁+l₂=100cm. At balance, resistances are in same ratio as wire lengths; swap to verify if l₁ → (100−l₁).
Series-Parallel Resistor Shortcut — For two resistors: Series = R₁+R₂; Parallel = (R₁×R₂)/(R₁+R₂). Remember "Product over Sum" for parallel only when n=2.
Kirchhoff's Current Law (KCL) — "Junction income = junction expense." Sum of currents entering = sum leaving any node; assign + to incoming, − to outgoing.
De Broglie Wavelength λ=h/p — For electron accelerated through V volts: λ = 12.27/√V Ångstroms. Remember "12.27" as dozen-and-quarter for quick numerical solving.
Photoelectric Work Function — KEmax = hν − φ. If frequency below threshold (hν < φ), zero electrons emitted regardless of intensity; intensity only affects photocurrent above threshold.
Chemistry
Quantum Number Limits — For n: l = 0 to (n−1); for l: m = −l to +l; s = ±½. Mnemonic: "Never Let Monkeys Scream" – hierarchy n→l→m→s.
Aufbau Order (1s to 4s) — Use (n+l) rule: fill lowest (n+l) first; if equal, lower n first. Quick: 1s 2s 2p 3s 3p 4s 3d 4p (4s fills before 3d because 4+0 < 3+2).
Hund's Rule Reminder — Electrons fill degenerate orbitals singly (parallel spins) before pairing. "Bus Seat Rule" – passengers occupy empty seats before sharing.
Molarity vs Molality — Molarity = moles/Liter (volume, temperature-dependent). Molality = moles/kg solvent (mass, temperature-independent). "Molarity has Measure (volume); molality has mass."
Mole Concept Bridge — 1 mole = 6.022×10²³ particles = 22.4L gas (STP) = molar mass in grams. Chain: mass → moles → molecules → volume (gas).
Stoichiometry Limiting Reagent — Calculate moles/coefficient for each reactant; smallest ratio = limiting reagent. Product yield calculated from limiting reagent only.
Carbohydrate Classification — Monosaccharides (glucose), Disaccharides (sucrose = glucose+fructose), Polysaccharides (starch, cellulose). Mnemonic: "MDP – Mono, Di, Poly."
DNA vs RNA — DNA: Deoxyribose, Thymine, Double helix. RNA: Ribose, Uracil, Single strand. "DNA is Double, Thymine; RNA is Ribose, Uracil."
Essential Amino Acids — "PVT TIM HALL" – Phenylalanine, Valine, Threonine, Tryptophan, Isoleucine, Methionine, Histidine, Arginine, Leucine, Lysine (10 essential for humans).
Fat-Soluble Vitamins — "ADEK" (A, D, E, K) are fat-soluble; rest (B-complex, C) are water-soluble. Fat-soluble stored in liver, can cause toxicity if excess.
VSEPR Geometry Quick — 2 pairs = linear; 3 = trigonal planar; 4 = tetrahedral; 5 = trigonal bipyramidal; 6 = octahedral. Subtract lone pairs for molecular shape (e.g., 4 pairs with 1 lone = pyramidal).
Hydrogen Bonding FON Rule — Strong H-bonds form when H attached to F, O, N (high electronegativity). "FONe" it in – fluorine, oxygen, nitrogen only.
Mathematics
Direction Ratios to Cosines — If direction ratios are (a,b,c), direction cosines = (a/√(a²+b²+c²), b/√(a²+b²+c²), c/√(a²+b²+c²)). Also: l²+m²+n²=1 always.
Shortest Distance Between Skew Lines — Formula: |[(b₁−b₂)·(a₁×a₂)]| / |a₁×a₂|. Mnemonic: "Vector connecting points, dot with cross of directions, divide by cross magnitude."
Plane Equation Forms — ax+by+cz=d (general); (a,b,c) is normal vector. Distance from (x₁,y₁,z₁) to plane = |ax₁+by₁+cz₁−d|/√(a²+b²+c²). Plug-and-chug with normal vector.
Binomial Coefficient Symmetry — ⁿCᵣ = ⁿCₙ₋ᵣ. Middle term(s): if n even, (n/2+1)th term; if n odd, ((n+1)/2)th and ((n+3)/2)th terms have max coefficient.
Greatest Coefficient Trick — In (1+x)ⁿ, coefficient maximum at middle term(s). For numerical max, check terms near (n+1)/(k+1) = x where k is term number.
Binomial rth Term — Tᵣ₊₁ = ⁿCᵣ · aⁿ⁻ʳ · bʳ (not Tᵣ!). Remember "+1 shift" – 3rd term means r=2. Independent of x: sum of powers of x and 1/x = 0.
Circle General Equation — x²+y²+2