Physics
Ohm's Law Triangle — Cover the unknown (V, I, or R) in a V=IR triangle. V on top, I and R below. Cover V to get I×R, cover I to get V/R, cover R to get V/I.
Right-Hand Thumb Rule — Thumb points in current direction, curled fingers show magnetic field direction around a straight conductor. For coils, fingers follow current, thumb points to North pole.
CIVIL for AC Circuits — In a Capacitor, I leads V; in an Inductor, V leads I. Helps remember phase relationships in AC circuits instantly.
Escape Velocity Formula — √(2GM/R) = √(2gR). Remember "2 grave R" where g·R appears. Same formula works for any celestial body; just plug values.
Lenz's Law Hand Trick — Induced current opposes the change. If flux increases inward, induced field points outward (and vice versa). Use right-hand rule with opposition in mind.
VIBGYOR for EM Spectrum — Violet Indigo Blue Green Yellow Orange Red for visible light order by increasing wavelength (decreasing frequency/energy).
Work-Energy Sign Rule — Work BY the system is positive (expansion), work ON the system is negative (compression). ΔU = Q - W where W is work by system.
Projectile Range Formula — Maximum range at 45°. For complementary angles (30°-60°, 20°-70°), range is identical. R = u²sin(2θ)/g.
Photoelectric KEY — Kinetic energy of electron = hf - Φ (work function). Frequency below threshold means zero emission, regardless of intensity.
Bohr Radius Scaling — rₙ ∝ n²/Z. First orbit (n=1) for hydrogen = 0.53 Å. For He⁺ (Z=2), divide by 2; for n=2, multiply by 4.
Fleming's Left Hand Rule — Thumb = Force/Motion, First finger = Field (B), Middle finger = Current (I). For motor effect where current-carrying conductor moves in magnetic field.
Capacitor Series/Parallel — Capacitors behave OPPOSITE to resistors: series → 1/C_total = Σ(1/C); parallel → C_total = ΣC. Remember "capacitors are contrary."
Chemistry
HOCN Rule for Isomerism — Chain, position, functional, metamerism, tautomerism possible in compounds with Heteroatoms, Oxygen groups, Carbon chains, Nitrogen groups.
Lucas Test Timing — 1° alcohol = no turbidity (or very slow); 2° alcohol = 5 minutes; 3° alcohol = immediate turbidity with ZnCl₂/HCl. "3-2-1 speed order."
LiNa KRb Cs Fr — "Little Naughty Kids Rub Costly Furniture" for Group 1 alkali metals in order of atomic number.
Be Mg Ca Sr Ba Ra — "Betty Manages Calcium Supplements By Radium" for Group 2 alkaline earth metals, increasing reactivity downward.
Aufbau Order Trick — Write 1s, 2s 2p, 3s 3p 3d, 4s 4p 4d 4f... diagonally. Fill arrows: 1s→2s→2p→3s→3p→4s→3d→4p→5s→4d... Each diagonal left-to-right.
SN1 vs SN2 — SN1: 3°>2°>1°, carbocation, racemization, polar prover. SN2: 1°>2°>3°, backside attack, inversion (Walden), aprotic solvent preferred.
Carbocation Stability — 3° > 2° > 1° > methyl. Hyperconjugation and resonance increase stability. More alkyl groups = more stable carbocation.
Aldehyde vs Ketone Tests — Tollens' (silver mirror), Fehling's (red ppt), both positive for aldehydes only. Ketones don't reduce these reagents (no oxidation).
Amine Basicity Order — In gas phase: 3°>2°>1°>NH₃. In water (aliphatic): 2°>1°>3°>NH₃. Aromatic amines weakest due to resonance electron delocalization into benzene.
Hinsberg's Test — 1° amine + benzene sulfonyl chloride → soluble in alkali; 2° amine → insoluble precipitate; 3° amine → no reaction. "1-dissolves, 2-drops, 3-does nothing."
Hybridization Shortcut — Count σ bonds + lone pairs on central atom: 2=sp, 3=sp², 4=sp³, 5=sp³d, 6=sp³d². Ignore π bonds entirely.
Mathematics
Divisibility by 11 — Alternating sum of digits. If (sum of odd-position digits) - (sum of even-position digits) is 0 or divisible by 11, number is divisible by 11.
Quadratic Roots Sum/Product — For ax²+bx+c=0: sum of roots = -b/a, product = c/a. Never re-solve; directly find α+β and αβ from coefficients.
nPr and nCr Quick Formula — nPr = n!/(n-r)!, nCr = n!/[r!(n-r)!] = nPr/r!. Remember: Permutation divides by (n-r)!, Combination divides by both r! and (n-r)!.
Integration by Parts: ILATE — Choose 'u' by priority: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. Higher priority becomes u, lower becomes dv.
Determinant Row/Column Trick — If two rows (or columns) are identical or proportional, determinant = 0. Adding k×(one row) to another row doesn't change determinant value.
Arithmetic Progression Sum — Sₙ = n/2[2a + (n-1)d] or n/2[first term + last term]. For finding middle term: T_middle = (a+l)/2 when n is odd.
sin²θ + cos²θ = 1 Family — Also 1+tan²θ = sec²θ and 1+cot²θ = csc²θ. Divide Pythagorean identity by cos²θ or sin²θ respectively to derive.
Slope-Intercept Forms — y = mx + c (slope m, y-intercept c). For point (x₁,y₁) and slope m: y - y₁ = m(x - x₁). Two points: m = (y₂-y₁)/(x₂-x₁).
Probability Addition Rule — P(A∪B) = P(A) + P(B) - P(A∩B). Subtract intersection to avoid double-counting. For mutually exclusive events, P(A∩B) = 0.
Derivative of Standard Functions — d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(cos x) = -sin x. Memorize these five cold.
Matrices Multiplication Compatibility — (m×n)×(n×p) = (m×p). Inner dimensions must match (both n); result has outer dimensions. Non-commutative: AB ≠ BA generally.
**Distance Formula