Mathematics
Sum of n terms A.P. — Use "S = N/2 × (A + L)" where N=number of terms, A=first term, L=last term. Alternative: S = N/2 × [2A + (N-1)D]. Remember "NAL" for quick recall.
Infinite G.P. sum — Sum = a/(1-r) only when |r| < 1. If common ratio magnitude ≥ 1, series diverges. Quick check: multiply first term by "1 over 1-minus-ratio."
Binomial middle term — For (a+b)ⁿ: if n is even, middle term is (n/2 + 1)th term. If n is odd, two middle terms are (n+1)/2 and (n+3)/2 positions.
De Moivre's Theorem — (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). Remember "powers multiply angles"—crucial for complex number powers and roots in polar form.
Determinant row-column trick — If any two rows/columns are identical or proportional, determinant = 0 instantly. Saves computation time; check before expanding.
nPr and nCr quick formula — nPr = n!/(n-r)!, nCr = n!/[r!(n-r)!]. Remember: Permutation divides once, Combination divides twice. Also nCr = nC(n-r).
Quadratic sum-product roots — For ax²+bx+c=0: sum of roots = -b/a, product = c/a. Mnemonic: "Suma-Minus-B, Produca-C" (both over a).
Logarithm base change — log_b(a) = log_k(a)/log_k(b) for any base k. Useful trick: log_a(b) × log_b(a) = 1 always.
Matrix inverse 2×2 shortcut — For [[a,b],[c,d]]: swap diagonal (a↔d), negate off-diagonal (b,c get minus), divide by determinant (ad-bc). Fast evaluation trick.
Standard Deviation quick — σ = √[Σ(x-mean)²/n]. Alternative: σ = √[(Σx²/n) - mean²]. Second formula often faster for calculation.
Physics
Ohm's Law triangle — Cover V to get V=IR, cover I to get I=V/R, cover R to get R=V/I. Draw triangle with V on top, I and R below.
Right-Hand thumb rules — Thumb=current/velocity, fingers=magnetic field, palm=force direction. For coil: curl fingers with current, thumb shows North pole. Two different contexts, both right-hand.
Kirchhoff's Laws memory — KCL: "Current Junction" (algebraic sum of currents at junction = 0). KVL: "Voltage Loop" (algebraic sum of voltages in loop = 0).
Photoelectric equation — KE_max = hν - φ. Remember "New minus Old": incident photon energy minus work function equals maximum kinetic energy of electron.
Newton's Gravitation — F = GMm/r². Remember g = GM/R² for surface. Mnemonic: "Gee-Emm over R-square" for acceleration due to gravity.
Kinematic equation selection — No distance? Use v=u+at. No time? Use v²=u²+2as. No final velocity? Use s=ut+½at². No acceleration? Use s=(u+v)t/2.
VIBGYOR wavelength — Violet shortest (~400nm), Red longest (~700nm). In glass, violet bends most. Remember: "Smaller wavelength, Bigger bend" for dispersion.
Bulk Modulus vs others — Bulk = volume stress/strain, Young's = length, Shear = shape. Volume-Bulk, Length-Young, Shape-Shear matching trick.
Chemistry
Electronegativity order — F > O > N > Cl. Mnemonic: "FONCl" (phonics). Fluorine most electronegative, follows oxygen-nitrogen-chlorine down.
Aldol condensation condition — Requires α-hydrogen (hydrogen on carbon adjacent to C=O). No α-hydrogen = no aldol. Check α-position first before attempting mechanism.
Carbocation stability — 3° > 2° > 1° > methyl. Mnemonic: "More substitution, More stable". Hyperconjugation and induction both increase with alkyl groups attached.
Lucas Test timing — Tertiary alcohol: cloudiness immediate. Secondary: 5 minutes. Primary: no reaction at room temp. Remember "3-2-1, Fast-Medium-Slow."
Coordination number patterns — Octahedral=6, Tetrahedral=4, Square planar=4. Common: [Co(NH₃)₆]³⁺ is 6, [Ni(CO)₄] is 4. Check geometry from coordination number.
SN1 vs SN2 quick — SN1: tertiary halides, polar solvent, two-step, racemization. SN2: primary halides, aprotic solvent, one-step, inversion. Remember "1=slow-racemic, 2=fast-flip."
Hess's Law shortcut — ΔH is state function: net enthalpy = sum of steps regardless of path. Reverse reaction flips sign, multiply equation multiplies ΔH proportionally.