Mathematics
BODMAS trap — Always do Division and Multiplication left-to-right (whichever comes first), same for Addition and Subtraction. Don't prioritize D over M or A over S; it's strictly left-to-right within each pair.
Divisibility by 11 — Alternating digit sum: subtract odd-position digits from even-position digits. If result is 0 or divisible by 11, the number is divisible by 11. Example: 121→(2)−(1+1)=0, divisible.
Quadratic sum/product — For ax²+bx+c=0, sum of roots = -b/a, product = c/a. Quick check: if roots are 2,3 then equation is x²−5x+6=0 (middle term negative of sum).
Arithmetic mean shortcut — For evenly-spaced progressions (AP), mean = (first term + last term)/2. Also equals the middle term if odd number of terms. Saves summing all terms.
Coordinate midpoint — Midpoint of (x₁,y₁) and (x₂,y₂) is ((x₁+x₂)/2, (y₁+y₂)/2). Section formula: for m:n ratio, x=(mx₂+nx₁)/(m+n). Internal division only; external uses minus.
Sin/Cos/Tan values — For 0°,30°,45°,60°,90°: sin = √0,√1,√2,√3,√4 all divided by 2; cos is reverse. Tan = sin/cos. Memorize the √0-to-√4 sequence for instant recall.
Pythagoras triples — Common sets: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Multiples also work: (6,8,10). Recognizing these saves calculation time in triangle problems.
Circle tangent property — Tangent is perpendicular to radius at point of contact (90° angle). Two tangents from external point are equal length. Angle between tangent and chord equals inscribed angle in alternate segment.
Probability = favorable/total — Always identify sample space first. For two independent events, P(A and B)=P(A)×P(B). For mutually exclusive, P(A or B)=P(A)+P(B). Deck has 52 cards, 4 suits.
Cylinder/Cone volume link — Cone volume = ⅓×base area×height. Cylinder = base area×height. Sphere = (4/3)πr³. Remember: cone is always one-third of corresponding cylinder with same base and height.
Physics
VIBGYOR wavelength — Violet shortest wavelength (~400nm), Red longest (~700nm). In denser medium, all bend toward normal except when incident perpendicular. Frequency never changes during refraction, only wavelength and speed.
Refractive index trick — n=c/v=(Real depth)/(Apparent depth)=sin i/sin r. For light going from rarer to denser, ray bends toward normal. Glass n≈1.5, water≈1.33, air≈1.0.
Newton's Laws mnemonic — First: Inertia (body resists change). Second: F=ma (force produces acceleration). Third: Action-Reaction pairs (equal, opposite, on different bodies). Always identify which body experiences which force.
Gravitational PE/KE — PE=mgh (near surface), KE=½mv². At maximum height, KE=0, PE=max. During free fall, PE converts to KE; total mechanical energy conserved (no air resistance).
Resistance combinations — Series: R_total=R₁+R₂+R₃ (resistances add). Parallel: 1/R_total=1/R₁+1/R₂+1/R₃ (reciprocals add). For two in parallel: R=(R₁×R₂)/(R₁+R₂). Current same in series, voltage same in parallel.
Right-hand thumb rule — Thumb points current direction, curled fingers show magnetic field direction around wire. For solenoid: fingers curl with current, thumb points to North pole. Fleming's rules for motor/generator effects.
Sign convention in mirrors — Distances measured from pole: toward object (left) is positive for object distance, toward right is positive for image distance in Cartesian convention. Focal length: concave negative, convex positive.
Specific heat vs Latent heat — Specific heat changes temperature (Q=mcΔT). Latent heat changes state at constant temperature (Q=mL). Water: c=4200 J/kg°C, L_fusion=336 kJ/kg, L_vaporization=2260 kJ/kg.
Chemistry
pH scale memory — pH 7=neutral, <7 acidic, >7 basic. Each unit is 10× change in H⁺ concentration. pH+pOH=14 at 25°C. Strong acids: HCl, H₂SO₄, HNO₃. Strong bases: NaOH, KOH.
Periodic table trends — Across period (left→right): atomic size decreases, electronegativity increases, metallic character decreases. Down group: size increases, ionization energy decreases. Noble gases (Group 18) are most stable.
OILRIG redox — Oxidation Is Loss (of electrons), Reduction Is Gain (of electrons). Oxidizing agent gets reduced, reducing agent gets oxidized. Increase in oxidation number = oxidation; decrease = reduction.
Valency quick recall — Group 1=1, Group 2=2, Group 13=3, Group 14=4, Group 15=3, Group 16=2, Group 17=1. Hydrogen=1, Carbon=4, Nitrogen=3, Oxygen=2. Transition metals have variable valency.
Balancing chemical equations — Start with most complex molecule. Balance metals first, then non-metals, then hydrogen and oxygen last. Check atoms of each element on both sides. Never change subscripts, only coefficients.
Homologous series pattern — Alkanes: CₙH₂ₙ₊₂ (single bonds). Alkenes: CₙH₂ₙ (one double bond). Alkynes: CₙH₂ₙ₋₂ (one triple bond). Alcohols: CₙH₂ₙ₊₁OH. Each successive member differs by CH₂, has similar properties.
Electrochemical series — Metals above hydrogen (K, Na, Ca, Mg, Al, Zn, Fe) displace H₂ from acids. More reactive metals displace less reactive ones from salt solutions. Reactivity decreases down the series.
Hardness of water — Temporary hardness: Ca(HCO₃)₂, Mg(HCO₃)₂—removed by boiling. Permanent hardness: CaSO₄, MgSO₄, CaCl₂—removed by washing soda (Na₂CO₃) or ion-exchange/permutit. Soap forms scum with hard water.