Physics
Resistor color code — *B*lack *B*rown *R*ed *O*range *Y*ellow *G*reen *B*lue *V*iolet *G*rey *W*hite (0-9). Mnemonic: "Bad Beer Rots Our Young Guts But Vodka Goes Well" for resistance values.
Right-hand thumb rule — Thumb = current direction, curled fingers = magnetic field direction around a wire. For force on conductor: First finger = Field, seCond = Current, thuMb = Motion/force (Fleming's left-hand rule).
Lens/mirror formula sign convention — All measurements from pole/optical center: distances with light direction are positive, against are negative. Real images are positive side, virtual negative. Concave mirror/convex lens have positive focal lengths.
EM spectrum frequency order — "Religious Men In Varanasi Use Xylophone Guitars Constantly" = Radio, Microwave, Infrared, Visible, UV, X-ray, Gamma (increasing frequency, decreasing wavelength).
Projectile maximum range — Launch angle of 45° gives maximum range on level ground. Range formula: R = u²sin(2θ)/g. Remember: sin(90°) = 1 is maximum, so 2θ = 90° means θ = 45°.
Thermodynamic processes — Isothermal: Temperature constant (PV = constant). Adiabatic: No heat exchange (PVᵞ = constant). Isobaric: Pressure constant. Isochoric: Volume constant. Remember "I-A-I-I" pattern.
Chemistry
Electronegativity order — F O N Cl are top four: Fluorine (4.0) > Oxygen (3.5) > Nitrogen (3.0) > Chlorine (3.0). Mnemonic: "Foxes Only Nibble Clovers" for most electronegative elements.
Amino acids classification — Essential amino acids: PVT TIM HALL = Phe, Val, Trp, Thr, Ile, Met, His, Arg, Leu, Lys. These must come from diet, body cannot synthesize.
d-block filling order — For transition elements, 4s fills before 3d, but 4s empties *first* during ionization. Example: Fe is [Ar]3d⁶4s², but Fe²⁺ is [Ar]3d⁶ (not 3d⁴4s²).
Periodic table groups — Group 1: Alkali, Group 2: Alkaline Earth, Group 17: Halogens ("salt-formers"), Group 18: Noble gases. Reactivity: Group 1 increases down, Group 17 decreases down.
SN1 vs SN2 mechanism — SN1: 1 step determines rate (unimolecular), carbocation intermediate, 3° > 2° > 1° order, racemization occurs. SN2: 2 species in rate step (bimolecular), one step, 1° > 2° > 3° order, inversion.
Chelate effect ligands — "Every Dumb Person Eats Bread" = En (ethylenediamine), DMG (dimethylglyoxime), Phen (phenanthroline), EDTA, Bipy (bipyridyl) are common chelating ligands forming multiple bonds.
Mathematics
Trigonometric values — For 0°, 30°, 45°, 60°, 90°: sin values are √0/2, √1/2, √2/2, √3/2, √4/2. Reverse for cos. Tan = sin/cos. Remember "0, 1, 2, 3, 4 under root, all over 2."
Integration by parts — ILATE rule priority: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. Choose first function (u) from left, second (dv) from right in formula ∫u·dv = u·v - ∫v·du.
Circle equation recognition — General form x² + y² + 2gx + 2fy + c = 0 has center (-g, -f) and radius √(g² + f² - c). If g² + f² < c, no real circle exists.
Definite integral properties — ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx (King property). For even function: ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx. For odd function: ∫₋ₐᵃ f(x)dx = 0.
Conic section eccentricity — Circle: e = 0, Ellipse: 0 < e < 1, Parabola: e = 1, Hyperbola: e > 1. Mnemonic: "Close Enough Perhaps Hyper" for increasing eccentricity order.
Matrix determinant shortcuts — If any row/column is all zeros, det = 0. If two rows/columns are identical or proportional, det = 0. Determinant of triangular matrix = product of diagonal elements.
L'Hospital's rule conditions — Apply only for 0/0 or ∞/∞ indeterminate forms. Differentiate numerator and denominator separately (not quotient rule!). Can apply repeatedly until determinate form appears.
Binomial coefficient symmetry — ⁿCᵣ = ⁿCₙ₋ᵣ (choosing r is same as leaving n-r). Also, ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's triangle addition property). Maximum coefficient at middle term(s).