Mathematics
BODMAS Trap Awareness — In simplification, many assume left-to-right for equal operators; remember: Division/Multiplication are *equal* priority (left-to-right), same for Addition/Subtraction. Example: 6÷2×3 = (6÷2)×3 = 9, not 1.
Binomial Middle Term — For (a+b)ⁿ, middle term position: if n is even, it's the (n/2 + 1)th term; if n is odd, two middle terms at positions (n+1)/2 and (n+3)/2.
Complex Modulus Squaring — For |z₁ ± z₂|², expand as |z₁|² + |z₂|² ± 2Re(z₁z̄₂). Saves time over full Cartesian expansion in multiple-choice questions.
De Moivre's Quick Powers — (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). For roots: n-th roots of unity are e^(2πik/n), k=0,1,...,n-1; sum to zero.
Eccentricity Recall: CHEP — Circle (e=0), Hyperbola (e>1), Ellipse (e<1), Parabola (e=1). Helps instantly identify conic from eccentricity value.
Definite Integration Symmetry — ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even; equals 0 if f is odd. Saves half the calculation time.
Differential Equation Order/Degree — Order = highest derivative present; Degree = power of highest-order derivative *after* clearing radicals/fractions. Check degree only when polynomial in derivatives.
Vector Addition Triangle Close — If vectors form a triangle, their sum is zero (→AB + →BC + →CA = 0). For polygon: vector sum around closed path vanishes.
Physics
Right-Hand Rules Bundle — (1) Thumb=current, fingers=field for straight wire. (2) Curl fingers=current loop, thumb=field direction. (3) Fleming's Right (generator): thuMb=Motion, First=Field, seCond=Current.
Capacitor Series/Parallel Flip — Capacitors in series: 1/C_total = Σ(1/Cᵢ) like resistors in *parallel*. Capacitors in parallel: C_total = ΣCᵢ like resistors in *series*. Opposite behavior to resistors.
EMF Induction Sign — Lenz's law: induced current opposes flux change. Quick check: increasing flux → induced field opposes; decreasing flux → induced field aids original.
Escape Velocity = √2 × Orbital — v_escape = √(2GM/R) = √2 × v_orbital. For Earth: ~11.2 km/s escape vs ~7.9 km/s orbital at surface.
AC Impedance Triangle — Z² = R² + (X_L - X_C)². Draw right triangle: base=R, height=(X_L - X_C), hypotenuse=Z. Phase angle φ = tan⁻¹[(X_L - X_C)/R].
Photoelectric: "Stop" Voltage — eV₀ = KE_max = hf - φ. Stopping potential V₀ is independent of intensity; intensity only affects photocurrent (number of electrons).
Kinetic Theory PV=⅓Nmv²_rms — Links macroscopic PV to microscopic motion. Quick rearrangement: v_rms = √(3RT/M) for ideal gas, or √(3P/ρ) using density.