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.