Mathematics
CAST Diagram for Trig Ratios — In quadrants I, II, III, IV respectively: All ratios positive, only Sin positive, only Tan positive, only Cos positive. Remember "Add Sugar To Coffee" clockwise from Q-I.
Circle Sector/Segment Areas — Sector area = (θ/360°) × πr²; Segment area = Sector area − Triangle area. For quick checks: 90° sector = ¼ circle, 180° = semicircle.
AP nth Term Shortcut — aₙ = a + (n−1)d. For sum: Sₙ = (n/2)[2a + (n−1)d] or (n/2)(first + last). Middle term of odd-numbered AP = average of all terms.
Discriminant Quick Check — For ax² + bx + c = 0: D = b² − 4ac. D > 0 → two real roots; D = 0 → one repeated root; D < 0 → no real roots.
Coordinate Distance Formula — Distance = √[(x₂−x₁)² + (y₂−y₁)²]. Section formula (internally): x = (mx₂+nx₁)/(m+n), y = (my₂+ny₁)/(m+n). Midpoint just uses m = n = 1.
Probability = Favorable/Total — For dice: total outcomes = 6ⁿ (n dice). For cards: 52 total, 26 red, 26 black, 13 per suit, 4 aces, 12 face cards.
Volume Formulas Chain — Cube = a³; Cuboid = lbh; Cylinder = πr²h; Cone = ⅓πr²h; Sphere = 4/3πr³; Hemisphere = 2/3πr³. Cone/Pyramid always have ⅓ factor.
Linear Equations Elimination — To eliminate x: multiply equations so x-coefficients match, then subtract. To eliminate y: match y-coefficients. Cross-multiplication: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁).
Statistics Mode-Median-Mean — Mode = most frequent value. Median = middle value when arranged. Mean = sum/count. For grouped data: use modal class, median class formulas with L + [(n/2−cf)/f]×h.
Physics
V = IR (Ohm's Law) — Voltage = Current × Resistance. Series: R_total = R₁+R₂+...; Parallel: 1/R_total = 1/R₁+1/R₂+.... Current same in series, voltage same in parallel.
Newton's Laws Sequence — I: Inertia (body stays at rest/motion unless force acts). II: F = ma. III: Action-Reaction equal and opposite. Remember "I-FAR": Inertia, Force-acceleration, Action-Reaction.
UVAST Equations — v = u + at; s = ut + ½at²; v² = u² + 2as; s_nth = u + a(n−½). For free fall: u=0, a=g=9.8 m/s².
Gravitational Formula — F = G(m₁m₂)/r². On Earth surface: g = GM/R² ≈ 9.8 m/s². Weight = mg; weight becomes zero at Earth's center (g=0 there).
Lens Formula — 1/v − 1/u = 1/f. Magnification m = v/u = h'/h. Convex lens: f positive (converging); Concave lens: f negative (diverging). Sign convention: distances measured from optical center.
Right-Hand Thumb Rule — Thumb = current direction, curled fingers = magnetic field direction around wire. For solenoid: fingers curl with current, thumb shows N-pole direction.
Power = Work/Time = VI — Electrical power P = V²/R = I²R. Energy = Power × Time. 1 kWh = 1 unit = 3.6 × 10⁶ J. Series bulbs: lower wattage glows brighter.
Work-Energy Theorem — Work = Force × Displacement × cosθ. KE = ½mv²; PE = mgh. Law of conservation: Total energy (KE + PE) remains constant in absence of friction.
Chemistry
pH Scale Memory — pH < 7 = Acidic, pH = 7 = Neutral, pH > 7 = Basic. Strong acids: HCl, H₂SO₄, HNO₃. Strong bases: NaOH, KOH, Ca(OH)₂.
Atomic Structure (2,8,8,18 Rule) — Shells K, L, M, N hold max 2n² electrons: K=2, L=8, M=18, N=32. Valence electrons = outermost shell electrons determine reactivity.
Valency Trick — Groups 1,2,3 have valency 1,2,3. Groups 15,16,17 have valency 3,2,1 (8−group number). Group 18 = 0 (noble gases, stable octet).
Reactivity Series — "Please Send Charlie's Monkeys And Zebras In Lead Cages Securely Guarded Properly." = K, Na, Ca, Mg, Al, Zn, Fe, Pb, H, Cu, Hg, Ag, Au, Pt. More reactive = displaces less reactive metal.
Periodic Table Trends — Atomic size decreases left→right (across period), increases top→bottom (down group). Metallic character decreases left→right. Non-metallic character increases left→right.
OILRIG for Redox — Oxidation Is Loss (of electrons), Reduction Is Gain (of electrons). Oxidizing agent gets reduced; Reducing agent gets oxidized.
Hydrocarbon Naming — Meth(1C), Eth(2C), Prop(3C), But(4C). Suffix: -ane (single bond), -ene (double bond), -yne (triple bond). Functional groups: -OH = alcohol, -COOH = carboxylic acid, -CHO = aldehyde.
Balancing Equations Method — Balance atoms appearing least first, leave H and O for last. Check: atom count same on both sides. Law of conservation of mass: total mass reactants = total mass products.