Mathematics
Divisibility by 3 or 9 — Sum all digits; if sum divides by 3, number divides by 3; if sum divides by 9, number divides by 9. Quick mental check for factors.
Quadratic roots sum & product — For ax² + bx + c = 0: sum of roots = −b/a, product = c/a. Saves time finding roots in MCQs without solving fully.
AP nth term shortcut — For arithmetic progression: an = a + (n−1)d. Common term between two APs: equate expressions, solve for n. Essential for progression problems.
Pythagorean triplets (3-4-5 family) — Memorize: 3-4-5, 5-12-13, 7-24-25, 8-15-17. Multiples also work (6-8-10, 9-12-15). Instantly solve right-triangle questions.
Statistics mode-median-mean — Mode = most frequent; Median = middle value (arrange first); Mean = sum/count. For grouped data: memorize modal class, median class formulas using frequency tables.
Probability = Favorable/Total — List all outcomes systematically (tree diagrams for compound events). P(A or B) = P(A) + P(B) − P(A and B). Cards: 52 total, 13 each suit.
Circle tangent property — Tangent perpendicular to radius at contact point. Two tangents from external point are equal length. Draw radius to solve tangent-length problems instantly.
Coordinate midpoint & distance — Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Distance: √[(x₂−x₁)² + (y₂−y₁)²]. Section formula for dividing line m:n internally: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
Trigonometry: SOHCAHTOA — Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Special angles: sin 30°=½, cos 60°=½, sin 45°=cos 45°=1/√2, tan 45°=1.
Polynomial factor theorem — If f(a) = 0, then (x−a) is a factor. Test small integers (±1, ±2, ±3) to find roots quickly for factorization questions.
Volume & surface area quick ratios — Cube: V=a³, TSA=6a². Cylinder: V=πr²h, CSA=2πrh. Sphere: V=(4/3)πr³, SA=4πr². Cone: V=(1/3)πr²h, CSA=πrl. Compare formulas to avoid mixing.
Linear equations elimination method — Multiply equations to make one variable's coefficient equal, then subtract. Faster than substitution for numerical coefficients in TS POLYCET.
Physics
Ohm's Law triangle (V-I-R) — Draw triangle: V on top, I and R at bottom. Cover what you need: V=IR, I=V/R, R=V/I. Works for power too (P-V-I).
Fleming's Left Hand Rule (motor) — Thumb = force/motion, Forefinger = magnetic Field, Middle finger = Current. For electromagnetic motor effect direction.
Concave mirror formula sign convention — 1/f = 1/v + 1/u. Object distance u always negative, focal length f negative for concave, positive for convex. Virtual image v is positive.
Refraction: "Fast to Slow bends Toward" — Light bends toward normal entering denser medium (air to glass), away from normal leaving (glass to air). Refractive index n = c/v = sin i/sin r.
Lens formula trick — 1/f = 1/v − 1/u (same as mirror but sign carefully). Power P = 1/f(meters), unit dioptre. Convex lens converges (f positive), concave diverges (f negative).
Resistance series & parallel — Series: R_total = R₁ + R₂ + R₃ (adds up). Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ (reciprocals add). Two equal resistors in parallel = half resistance.
Heat transfer H = mc∆T — Heat gained/lost = mass × specific heat × temperature change. Units: calories or joules. Latent heat L for phase change: H = mL (no temperature change).
Eye defects correction — Myopia (short sight): distant objects blurred, use concave lens. Hypermetropia (long sight): near objects blurred, use convex lens. Presbyopia (aging): bifocals.
Chemistry
pH scale quick values — pH < 7 acidic, pH = 7 neutral, pH > 7 basic. Strong acids (HCl, H₂SO₄) pH~1-2. Strong bases (NaOH) pH~13-14. Memorize common indicators: litmus, phenolphthalein, methyl orange.
Atomic structure: 2n² electrons — Maximum electrons per shell = 2n². K shell (n=1): 2, L shell (n=2): 8, M shell (n=3): 18. Valence electrons in outermost shell determine reactivity.
Periodic table trends — Across period (left to right): atomic size decreases, electronegativity increases, metallic character decreases. Down group: atomic size increases, reactivity varies, valence electrons same.
Valency from group number — Groups 1-4: valency = group number. Groups 5-7: valency = 8 − group number. Group 8 (noble gases): valency = 0. Quick formula writing.
Balancing equations atom-count — Count each element's atoms on both sides. Start with most complex molecule. Balance metals first, then non-metals, oxygen and hydrogen last. Check final tally.
Carbon bonding tetravalency — Carbon forms 4 bonds (valency 4), allows chains, branches, rings. Saturated (single bonds): alkanes (C_nH_{2n+2}). Unsaturated: alkenes (C_nH_{2n}), alkynes (C_nH_{2n−2}).
Reactivity series mnemonic — "Please Send Charlie's Monkeys And Zebras In Lead Cages (Heavy) Securely Guarded" = K, Na, Ca, Mg, Al, Zn, Fe, Pb, Cu, Hg, Ag, Au. Higher displaces lower from salt solution.
Biology
Cell organelles functions — Mitochondria: powerhouse (ATP). Ribosomes: protein synthesis. ER: transport. Golgi: packaging. Lysosomes: digestion. Chloroplasts (plants): photosynthesis. Nucleus: control center/genetic material.
Five-kingdom classification — Monera (bacteria), Protista (amoeba), Fungi (mushroom), Plantae (ferns to flowering), Animalia (sponges to mammals). Based on cell structure, nutrition, body organization.
Photosynthesis equation — 6CO₂ + 6H₂O + light energy → C₆H₁₂O₆ + 6O₂ (chlorophyll catalyst). Occurs in chloroplasts. Raw materials: CO₂ and water; products: glucose and oxygen.
Mendel's Law shortcut — Dominant (TT, Tt) masks recessive (tt). Monohybrid F₂ ratio: 3:1 phenotype, 1:2:1 genotype. Dihybrid F₂: 9:3:3:1. Punnet square for crosses.
Human excretory system pathway — Kidney (nephron filters blood) → Ureter (carries urine) → Urinary bladder (stores) → Urethra (expels). Nephron unit: glomerulus