Mathematics
Binomial Coefficient Sum — Sum of binomial coefficients: Set x=1 in (1+x)ⁿ to get 2ⁿ; for alternating sum set x=−1 to get 0. Quick check for expansion questions.
Circle Equation Toggle — General form x²+y²+2gx+2fy+c=0 has center (−g,−f) and radius √(g²+f²−c). Remember "negative of half-coefficients" for center coordinates.
Complex Modulus–Argument — |z₁z₂|=|z₁||z₂| and arg(z₁z₂)=arg(z₁)+arg(z₂). Product means multiply moduli, add arguments; division means divide moduli, subtract arguments. Essential for De Moivre problems.
Eccentricity Ladder — Circle e=0, Ellipse e<1, Parabola e=1, Hyperbola e>1. Remember "CEP-H" (0,<1,1,>1) for increasing eccentricity order in conic sections.
Integration By Parts — ILATE — Choose 'u' in order: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. First in list becomes u, rest becomes dv for ∫u·dv formula.
Derivative of aˣ — d/dx(aˣ) = aˣ·ln(a). Many students forget the ln(a) multiplier; remember "exponential brings natural log of base."
Direction Cosines Sum — For direction cosines l,m,n: l²+m²+n²=1 always. Quick verification tool for 3D geometry answer-checking in MCQs.
Physics
Ohm's Law Triangle — Draw triangle: V at top, I and R at bottom. Cover what you want: V=IR, I=V/R, R=V/I. Works for all variant formulas.
Right-Hand Thumb Rule — Thumb=current direction, curled fingers=magnetic field direction. For force: First finger=Field, seCond=Current, thuMb=Motion (Fleming's left-hand rule for motors).
Kirchhoff's Current Law — Junction — "Current IN = Current OUT" at any junction. Algebraic sum of currents at a node equals zero; assign signs carefully (enter +ve, exit −ve).
Capacitor Series–Parallel — Series: 1/C_total = 1/C₁+1/C₂ (like resistors in parallel). Parallel: C_total = C₁+C₂ (like resistors in series). Opposite of resistor rules.
Lenz's Law — Oppose — Induced current's magnetic field opposes the change causing it. Think "nature is lazy, resists change." Essential for electromagnetic induction direction problems.
Photoelectric Work Function — Einstein equation: KE_max = hν − φ. Threshold frequency: hν₀=φ. Below threshold, no emission regardless of intensity. Remember intensity affects number, not energy.
Chemistry
Acidity Order — Phenols — Electron-withdrawing groups (−NO₂, −Cl) increase acidity; electron-donating (−CH₃, −OCH₃) decrease it. Ortho/para positions have stronger effect than meta.
— Tertiary alcohols react (cloudiness), secondary in , primary at room temperature with Lucas reagent (ZnCl₂+HCl).