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).