Physics
Fleming's Left & Right Hand Rules — LEFT for motor (Force), RIGHT for generator (Induced current). Thumb=Force/Motion, First finger=Field, seCond finger=Current. Remember: "moLor Left, geneRatoR Right."
Amplitude Modulation Band Math — For AM radio, bandwidth = 2×message frequency. Side bands at fc±fm. Modulation index μ = Am/Ac; μ≤1 prevents distortion. Power in side bands = (μ²/2)Pc.
Series vs Parallel Resistance/Capacitance Flip — Resistors in series ADD; capacitors in series use reciprocal sum (1/Ceq = 1/C₁+1/C₂). In parallel, it flips: resistors use reciprocal, capacitors ADD directly. Opposite behaviors, easy to confuse.
Photoelectric Equation Anchor — KEmax = hν - φ (work function). Stopping potential: eV₀ = KEmax. Below threshold frequency ν₀ = φ/h, NO emission regardless of intensity. Intensity affects current, NOT electron energy.
de Broglie Wavelength Shortcuts — λ = h/p = h/mv. For electrons accelerated through V volts: λ = 12.27/√V Ångströms (derive from ½mv² = eV). Heavier particle = shorter wavelength at same speed.
Bohr Radius & Energy Series — rn = 0.529n²/Z Ångströms (n=1,2,3...). En = -13.6Z²/n² eV. For hydrogen Z=1: r₁=0.529Å, E₁=-13.6eV. Energy released in transition: ΔE = 13.6(1/n₁² - 1/n₂²) eV.
Self-Inductance Flux Rule — ε = -L(dI/dt). Energy stored = ½LI². For solenoid L = μ₀n²Al (n=turns/length, A=area). Inductor opposes CHANGE in current, acts like inertia for electricity.
Escape Velocity Quick Formula — ve = √(2gR) = 11.2 km/s for Earth (g=10 m/s², R=6400 km). Independent of projectile mass. Derive from KE = PE: ½mv² = GMm/R. Orbital velocity = ve/√2.
Chemistry
Lucas Test Timing — 1° alcohols: no turbidity at room temp. 2° alcohols: turbidity in 5 minutes. 3° alcohols: immediate turbidity (ZnCl₂/HCl). Remember "1-Never, 2-Five, 3-Instant" for alcohol classification.
Aldehyde vs Ketone Tests — Aldehydes: Tollen's (silver mirror), Fehling's (red ppt), both POSITIVE. Ketones: NEGATIVE for both (except α-hydroxy ketones). Mnemonic: "AldeHYDE seeks SILVER lining."
Carbylamine Reaction — Primary amines ONLY react with CHCl₃ + alc. KOH to give foul-smelling isocyanide (RNC). 2° and 3° amines: NO reaction. Diagnostic test: "Carbylamine = 1° amine."
Hinsberg Reagent Pattern — Benzenesulfonyl chloride (C₆H₅SO₂Cl): 1° amine gives soluble product in KOH, 2° gives insoluble ppt, 3° doesn't react. "One dissolves, Two precipitates, Three stays away."
Azimuthal Quantum Number Shapes — l=0(s, sphere), l=1(p, dumbbell), l=2(d, double-dumbbell/cloverleaf), l=3(f, complex). For nth shell, l = 0 to (n-1). Remember: Spherical Peanut Double Flowery.
d-Block Magnetic Moment — μ = √[n(n+2)] BM, where n = unpaired electrons. Quick check: Ti³⁺(d¹)→√3, V³⁺(d²)→√8, Cr³⁺(d³)→√15, Fe²⁺(d⁶)→√24, Cu²⁺(d⁹)→√3 BM.
VBT Hybridization Counting — Add: ½(V + M - C + A). V=valence e⁻ of central atom, M=monovalent atoms, C=cation charge, A=anion charge. Result: 2=sp, 3=sp², 4=sp³, 5=sp³d, 6=sp³d².
Mole Concept STP Constants — At STP: 1 mole = 22.4 L gas = 6.022×10²³ particles = molar mass in grams. Quick check: 32g O₂ = 1 mole = 22.4L at STP.
Coordination Number-Geometry Map — CN 2: linear, CN 4: square planar (d⁸) or tetrahedral (d⁰/d¹⁰), CN 6: octahedral. [Ni(CN)₄]²⁻ square planar, [NiCl₄]²⁻ tetrahedral. Strong field → pairing → square planar.
Chemistry (continued)
Williamson Ether Synthesis Direction — RO⁻ + R'X → ROR' (SN2). Use less hindered alkyl for halide (R'X), more hindered for alkoxide (RO⁻). Best: 1° halide with any alkoxide. Prevents elimination side reactions.
Claisen vs Aldol — Aldol: aldehydes/ketones with α-H under base, forms β-hydroxy carbonyl. Claisen: esters with α-H + base, forms β-ketoester. Mnemonic: "Aldol needs Aldehyde H; Claisen needs ester C."
Mathematics
nCr Symmetry & Addition — ⁿCr = ⁿCn-r (choosing r = choosing remaining). ⁿCr + ⁿCr-1 = n+1Cr (Pascal's triangle). For max: nCn/2 when n even, nC(n±1)/2 when n odd.
Binomial Middle Term Trick — If n even: one middle term T(n/2)+1. If n odd: two middle terms T(n+1)/2 and T(n+3)/2. General term: Tr+1 = ⁿCr·xⁿ⁻r·yr (r starts from 0).
Divisibility by 9 & 11 — Div by 9: sum all digits divisible by 9. Div by 11: (sum odd positions - sum even positions) divisible by 11. Example: 1243: (1+4)-(2+3)=0, divisible by 11.
L'Hôpital 0/0 Forms — For lim(x→a) f(x)/g(x) giving 0/0 or ∞/∞: differentiate numerator and denominator separately. Repeat if needed. Works ONLY for indeterminate forms, not 1/0.
Integration by Parts: ILATE — Priority for choosing 'u': Inverse trig, Logarithm, Algebraic, Trig, Exponential. Higher priority becomes u, rest becomes dv. Formula: ∫u dv = uv - ∫v du.
Differential Equation Order/Degree — Order = highest derivative present. Degree = power of highest