Mathematics
A.P. Sum Formula Choice — If *n* is odd, use Sₙ = n × middle term. If *n* is even, use Sₙ = (n/2)[2a + (n−1)d]. Odd = direct middle; even = formula.
Circle Tangent Length — From external point, two tangents to a circle are *always equal in length*. If solving for x, set tangent segments equal immediately.
Coordinate Distance Shortcut — For distance between (x₁,y₁) and (x₂,y₂): square the differences, add, then root. Remember "Diff-Square-Add-Root" (DSAR).
Heron's Formula Memory — Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. Mnemonic: "Semi-perimeter Subtracted Successively" for the three (s−side) terms.
Sector Area Quick — Area of sector = (θ/360) × πr². Length of arc = (θ/360) × 2πr. Both start with (θ/360); area adds one *r*, arc adds none.
Tangent-Chord Angle — Angle between tangent and chord equals angle in alternate segment. Look for this in circle proofs—instant angle equality.
Parallelogram Area Trap — Area = base × perpendicular height, NOT base × slant side. Always drop perpendicular; never use the sloping edge directly.
Trigonometry Complementary — sin(90−θ) = cosθ; tan(90−θ) = cotθ; sec(90−θ) = cosecθ. "Complements switch to Co-functions."
Section Formula Direction — For internal division m:n, coordinates = (mx₂+nx₁)/(m+n). Heavier weight (m) pulls toward second point (x₂). Weight goes with *opposite* point.
Construction Circle Tangent — To draw tangent from external point: join to center, bisect that line, draw semicircle—intersections give tangent contact points. Semicircle method never fails.
Physics
Ohm's Law Triangle — Draw triangle: V on top, I and R at base. Cover what you need: V = IR, I = V/R, R = V/I. "Very Important Rule."
Fleming's Left-Hand Rule — Thumb = motion/Force, First finger = Field (N→S), Middle = Current direction. "Thumb Force, First Field, Middle Current" (ThuFF-MC for motor).
Right-Hand Thumb Rule — Thumb = current direction; curled fingers = magnetic field direction around straight wire. For coils, fingers = current, thumb = North pole inside.
Lens Formula Sign — Concave lens: f is negative. Object left of lens: u is negative. Real image right of lens: v is positive. "Left and inward = negative."
Power of Lens — P (diopter) = 1/f (meters). Concave has negative power; convex has positive power. Shorter focal length = more powerful lens. P₁+P₂ = combined power.
Equations of Motion Memory — v = u + at; s = ut + ½at²; v² = u² + 2as. Mnemonic "Very Useful Acceleration Tricks" for first letters.
Gravitational g vs G — Small *g* = 9.8 m/s² (Earth surface only). Big *G* = 6.67×10⁻¹¹ (universal constant). *g* changes with height; *G* never changes.
Resistance Series-Parallel — Series: Rₜₒₜₐₗ = R₁+R₂ (resistances add). Parallel: 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ (reciprocals add). Series = simple sum; parallel = reciprocal sum.
Speed of Sound Memory — In air ≈ 340 m/s (roughly 1 km in 3 seconds). Lightning-thunder gap: divide seconds by 3 to get distance in km.
Work-Energy Sign Rule — Work done *against* friction/gravity = negative. Work by applied force in motion direction = positive. Against = negative; along = positive.
Chemistry
Reactivity Series — "Please Send Charlie's Monkeys And Zebras In Large Cages Having Strong Guards" = K, Na, Ca, Mg, Al, Zn, Fe, Pb, H, Cu, Hg, Ag, Au, Pt (decreasing reactivity).
pH Scale Shortcuts — pH < 7 = acidic; pH = 7 = neutral; pH > 7 = basic. Strong acid pH ≈ 1–2; strong base pH ≈ 13–14. Each unit = 10× concentration change.
Valency of First 20 — H=1, He=0; C=4, N=3, O=2, F=1; Na=1, Mg=2, Al=3, Si=4, P=3/5, S=2, Cl=1, K=1, Ca=2. "He Hesitates, Carbon Never Offers Free" for 2–10.
Balancing Redox Shortcut — Oxidation = electrons lost (OIL). Reduction = electrons gained (RIG). "OIL RIG" for electron transfer direction.
Homologous Series CₙH₂ₙ₊₂ — Alkanes differ by CH₂. General formula CₙH₂ₙ₊₂. Methane (C₁), Ethane (C₂), Propane (C₃), Butane (C₄). "Me Eat Peanut Butter."
Matter State Changes — Solid→Liquid = melting; Liquid→Gas = vaporization; Solid→Gas direct = sublimation. Reverse: freezing, condensation, deposition. Sublimation skips liquid entirely (dry ice, iodine).
Periodic Table Groups — Group 1 = alkali metals; Group 2 = alkaline earth; Groups 3–12 = transition; Group 17 = halogens; Group 18 = noble gases (inert/stable).
Mole Concept Bridge — 1 mole = 6.022×10²³ particles = gram atomic/molecular mass. Avogadro's number bridges atoms to grams. Mass/Molar mass = moles.