Mathematics
Divisibility by 11 — Alternating sum: subtract odd-position digits from even-position digits; if result is 0 or divisible by 11, the number is too. Example: 121→(1+1)−2=0, divisible.
AP sum formula choice — If last term given, use Sn = n/2(a+l); if common difference given, use Sn = n/2[2a+(n−1)d]. Saves time in identifying which formula fits faster.
Circle angle theorems — "Arc double": angle at center = 2×angle at circumference on same arc. "Semicircle = 90°": angle in semicircle always right angle. Cyclic quad opposite angles sum to 180°.
Quadratic quick roots — For x²+(a+b)x+ab=0, roots are −a and −b. For ax²+bx+c, if a+b+c=0 then roots are 1 and c/a; if a−b+c=0 then roots are −1 and −c/a.
Distance-midpoint combo — Distance = √[(x₂−x₁)²+(y₂−y₁)²]; Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). Remember "half for midpoint, square-root for distance."
Cylinder-cone-sphere volume — Mnemonic "1:1/3:2/3 ratio" — Cylinder πr²h : Cone πr²h/3 : Hemisphere 2πr³/3. All share πr² base factor.
Heron's formula shortcut — s = (a+b+c)/2, Area = √[s(s−a)(s−b)(s−c)]. When sides are consecutive integers, s cancels nicely; check mental math first.
Probability "1 minus" — P(at least one) = 1 − P(none). P(exactly one) = P(A only)+P(B only). Convert tough "at least" problems to easier complements.
Physics
SUVAT equations memory — v=u+at, s=ut+½at², v²=u²+2as. Mnemonic: "VUAt, SUt-half, V²U²2AS" — links letters to formula structure for quick recall.
Right-hand thumb rules — Current direction: thumb=current, fingers curl=magnetic field (straight wire). For coil: curl fingers with current, thumb=North pole. Fleming's right for generator, left for motor.
Lens formula triangle — 1/v − 1/u = 1/f and m = v/u = h₂/h₁. Sign convention: real is positive side (for v,u), virtual negative. Convex lens f positive, concave negative.
Ohm's law extensions — Series: R_total = R₁+R₂, same current. Parallel: 1/R = 1/R₁+1/R₂, same voltage. "Series=Sum, Parallel=Product/Sum" for two resistors: R₁R₂/(R₁+R₂).
Work-energy-power chain — Work = Force×displacement×cosθ. KE = ½mv². Power = Work/time = Force×velocity. Remember: "Power needs velocity, Work needs displacement."
Gravity formulas cluster — g = GM/R², g_height = g(R/(R+h))², g_depth = g(1−d/R). Surface value 9.8 m/s²; doubles as acceleration for free fall calculations.
Reflection/Refraction laws — Reflection: i=r, same medium. Refraction: n=sin i/sin r = v₁/v₂ = λ₁/λ₂. "Rarer to denser bends toward normal, denser to rarer bends away."
Chemistry
pH scale memory — pH<7 acidic, pH=7 neutral, pH>7 basic. Strong acids (HCl, H₂SO₄, HNO₃) → "HiNdS" mnemonic. Each pH unit = 10× H⁺ concentration change.