Formula & Shortcut Sheets

Write dimensional grid [Total Top / Total Bottom] first: for equal-distance average speed use harmonic mean 2v1*v2/(v1+v2); for prefix conversions square/cube the power of 10 immediately (1 cm² = 10⁻⁴ m², 1 cm³ = 10⁻⁶ m³).

Factor out common surds before rationalising; memorise (a + b*sqrt(n))(a - b*sqrt(n)) = a² - n*b² and convert all base powers (4^x, 8^x, 9^x) to prime bases (2²x, 2³x, 3²x) in Line 1.

For rounding to unit u: [x - u/2, x + u/2); for truncation to unit u: [x, x + u). For extreme bounds of A/B or A-B, always cross-pair: Max = UB(A)/LB(B) and UB(A)-LB(B).

Never solve for proportionality constant k: write y2/y1 = (x2/x1)^n directly using fractional multipliers (e.g. +20% = 6/5, -25% = 3/4). For reverse percentages, divide by the multiplier.

Lock onto the invariant quantity (pure solute or pure metal) that does not change during dilution: C1*V1 = C2*V2. For two mixtures C1 and C2 making C_m, the volume ratio is |C2 - C_m| : |C_m - C1|.

Write the scaling triad L : A : V = k : k² : k³ immediately. For a frustum cut at fraction k of height from apex, V_frustum = (1 - k³) * V_whole.

For y = (ax+b)/(cx+d), the inverse rearrangement swaps (a, d) with a sign flip: x = (-dy+b)/(cy-a). For p(x+q)²+r, evaluate at x=0 and x=-q to find r in 10 seconds without full expansion.

Check if the target expression (e.g. 6a+2c) is a direct linear combination of the given equations (2*(4a+4c) - (2a+3c)) before solving for individual variables. For ax+by=c, gradient is -a/b.

Draw auxiliary radii to every tangent contact point (90 deg) and chord endpoint (isosceles triangle) first. Eliminate options by matching the pi term vs. the surd/rational triangle term separately.

Cancel pi on both sides before substituting numbers when equating cylinder/cone/sphere volumes. For 3D space diagonals, use d = sqrt(x² + y² + z²) in one step.

Never multiply large means: balance deviations from the target mean so sum(x_i - mean) = 0. In histograms, Area = Frequency (F = FD * width).

For 'at least one', always compute 1 - P(none). For 2-Group overlaps without formal set notation, use Total - Neither = Group1 + Group2 - Both.

Always work with exterior angles (360°/n): each clockwise vertex turn on a regular n-gon adds +360°/n to the bearing (mod 360°). On a clock at H:M, angle = |30H - 5.5M|°.

Never perform long division: write P(x) = Q(x)(x-a)(x-b) + mx + n, substitute x = a and x = b to get two instant linear equations ma + n = P(a) and mb + n = P(b).

In ESAT/TMUA, sqrt(u) is strictly >= 0. Before squaring sqrt(f(x)) = g(x), write the mandatory filter g(x) >= 0 on your notepad and discard any root that fails it.

Use the mean identities directly: three terms x, y, z are in AP iff 2y = x + z, and in GP iff y² = xz. Always verify |r| < 1 if S_infinity is mentioned.

For recurrence relations u_{n+1} = f(u_n) asking for u₁₀₀ or sum_{n=1}^{100}, compute u₁, u₂, u₃, u₄ first—90% of ESAT/TMUA recurrences are periodic with period 2, 3, 4, or 6. In sum_{r=m}^n, count = n - m + 1.

Never substitute y = mx + c into a circle if you can use geometry: a line is tangent iff perpendicular distance from centre (a,b) equals radius r, and shortest distance to an external point P is |CP| - r.

To count roots of trig(kx + a) = c in [0, 2*pi], never solve for x: substitute u = kx + a, transform the interval to [a, 2k*pi + a], and count wave intersections per 2*pi cycle.

Spot disguised quadratics immediately: 4^x = (2^x)² and 2^(x+k) = 2^k * (2^x). By Vieta's formulas, if u₁*u₂ = a^(x₁ + x₂) = c/A, then x₁ + x₂ = log_a(c/A) directly without finding x₁ or x₂!

ESAT/TMUA restricts differentiation to rational powers x^n: always split fractions over a single-term denominator x^k*sqrt(x) into separate powers x^(a - k - 1/2) first, then apply n*x^(n-1) term-by-term.

A cubic ax³+bx²+cx+d (a>0) is increasing for all x iff its derivative 3ax²+2bx+c >= 0 everywhere, which reduces to discriminant (2b)² - 4(3a)(c) <= 0 => b² <= 3ac.

For the area enclosed between a parabola y = ax² + bx + c and a line intersecting at x = alpha and x = beta, use Archimedes' 1/6 shortcut: Area = |a|/6 * (beta - alpha)³ in 10 seconds!

Over symmetric limits [-a, a], cross out all odd powers (x, x³, x⁵) immediately (= 0) and double the integral of even powers from [0, a]. For vertical shift +c, add rectangle area c*(b-a).

Track a single landmark point (vertex or intercept) through the equation: if (x₀, y₀) is on y = f(x), solve bx_new + c = x₀ for x_new and set y_new = a*y₀ + d.

Never expand the full bracket: write the general term T_r = nCr * a^(n-r) * b^r * x^(p(n-r) + qr), set the exponent p(n-r) + qr equal to the target power (0 for constant term), solve for r in 5 seconds, and evaluate only that single term.

Ambiguous Case (SSA) condition for 2 non-congruent triangles given angle A, adjacent side c, and opposite side a: require c * sin(A) < a < c, giving roots c*cos(A) ± sqrt(a² - c²*sin²(A)).

For ax² + bx + c = 0, the positive difference between roots is ALWAYS |alpha - beta| = sqrt(b² - 4ac) / |a|. For two distinct positive roots, intersect three conditions: Δ > 0, -b/a > 0, and c/a > 0.

Use the product-over-sum shortcut R_p = (R1*R2)/(R1+R2) (and R/n for n identical parallel resistors). In meter trend tables, take the extreme limit R_var -> infinity (open circuit) to read ammeter/voltmeter changes in 5 seconds.

Match the formula to the invariant: use P = I²*R for series components and transmission lines (same I); use P = V²/R for parallel branches connected across a fixed supply (same V).

In F = BIL = B(Q/t)L, L is strictly the perpendicular segment inside the magnetic field (in metres) and t must be in seconds. Parallel current wires attract; antiparallel wires repel.

Never use algebraic SUVAT if a v-t sketch gives a triangle/trapezium: displacement is simply average velocity (u+v)/2 * t. Always use g = 10 N/kg for vertical free-fall.

Two-line coupled-body rule: Line 1 (Whole System) a = F_net_total / (m1 + m2). Line 2 (Trailing Body only) T - F_drag_2 = m2 * a. If resistive force is proportional to mass, tension split is purely T = F_drive * m_rear / (m1 + m2).

In static or dynamic equilibrium (at rest or constant terminal velocity), net force in every direction is zero: upward tension/drag balances downward weight (T = mg or F_drag = mg), and pulley rope tension is uniform throughout a frictionless massless cord.

Spring stiffness behaves OPPOSITE to resistors: two identical springs in parallel are stiffer (k_eff = 2k, half the extension); in series they are softer (k_eff = k/2, double the extension). Always compute x = |L - L₀| first.

Draw a (+) arrow on the page before writing a single number: rebounding off a wall ALWAYS gives |Δp| = m(|v| + |u|), and kinetic energy relates to momentum via E_k = p² / (2m).

Write the Master Energy Accounting line: E_k_initial + mgh_initial = E_k_final + mgh_final + F_drag * d (with g = 10 N/kg). For constant velocity v against drag F, Mechanical Power is simply P = F * v.

Check mass units first: if c is in J/(kg °C), convert grams -> kg (÷1000) immediately. On a ΔT vs t graph with constant heater power P, gradient k = ΔT/t = P/(Mc).

Refraction invariant: frequency f NEVER changes across a boundary (v1/λ1 = v2/λ2). For echo/ultrasound, halve time (t/2) immediately; for a particle on a transverse wave, distance per cycle = 4 * Amplitude.

Decay chain 2-step rule: (1) Mass number A only changes via alpha: N_alpha = (A_start - A_end)/4. (2) Proton number: N_beta = Z_end - (Z_start - 2*N_alpha). For count rates: Net = Raw - Background.

For any magnet-coil motion graph: (1) Induced V = 0 whenever magnet velocity v = 0; (2) V flips sign on return journey; (3) Faster return = taller, narrower peak (same area ∫V dt = ΔΦ).

Isothermal gas shortcut: since P*V = const and ρ = m/V, gas density ρ scales directly with absolute pressure: ρ_2 / ρ_1 = P₂ / P₁ = (P_atm + mg/A) / P_atm.

Thermal Conduction Scaling Law: Rate Q̇ ∝ A * ΔT / L ∝ d² * (T2 - T1) / L. Radiation Law: Dull/black = best emitter & absorber; Shiny/white = worst emitter & best reflector.

For two isotopes of mass M1 and M2 with average A_r, use the lever rule: % of M2 = (A_r - M1)/(M2 - M1) * 100%. Isoelectronic species (N³-, O²-, F^-, Ne, Na^+, Mg²+) all share 2,8 (10 e^-).

Diatomic mass spectra follow (p + q)² = p² : 2pq : q². Memorise the two canonical ESAT ratios: Br2+ (1:1 isotopes at 79,81) -> 1 : 2 : 1 at m/z 158, 160, 162; Cl2+ (3:1 isotopes at 35,37) -> 9 : 6 : 1 at m/z 70, 72, 74.

By Avogadro's Law, gas volume ratios equal mole ratios directly: for 1 volume of C_xH_y burned, x = V(CO2)/V(alkane) immediately, and O2 consumed is (x + y/4) volumes. At rtp, H2O(l) volume = 0.

Memorise: 1 mol gas at rtp = 24 dm³ = 24,000 cm³, so 1 mmol gas = 24 cm³. Atom economy depends purely on the balanced equation (desired M_r / total reactant M_r), independent of experimental yield.

For disproportionation, look for a free element (e.g. Cl2, Br2, Cu+) or intermediate oxide (NO2, H2O2) on the LEFT that appears in TWO different products on the RIGHT with one higher and one lower oxidation state.

In structure tables, scan the Conductivity columns first: (Solid=No, Liquid=Yes) -> Giant Ionic; (Solid=Yes, Liquid=Yes) -> Metallic; (High MP, Solid=Yes) -> Graphite; (High MP, Solid=No, Liquid=No) -> SiO2/Diamond.

Check the exact wording on the mass given: if given M grams of WATER, Precipitate = (M/100)*(S_hot - S_cold). If given M grams of SATURATED SOLUTION at T_hot, Precipitate = M * (S_hot - S_cold) / (100 + S_hot).

Every 10-fold increase in total volume V_final / V_initial = 10^k increases strong acid pH by +k units (capped at pH 7). To shift pH by +k, you must add (10^k - 1)*V_initial of water.

Decompose any rate graph into two independent numbers: (1) Initial Slope at t=0 = Concentration / Temp / Catalyst / Surface Area; (2) Final Plateau Height = Moles of Limiting Reactant (n = cV).

For bond energies: ΔH = Broken (Left) - Formed (Right); cancel any bonds that remain unchanged on both sides before multiplying. In q = mcΔT, m is ALWAYS the mass of water/total aqueous solution being heated.

Aqueous discharge rules in 10s: Cathode (-) gives H2 unless metal is Cu, Ag, or Au. Anode (+) gives Halogen (Cl2, Br2, I2) if halide present, otherwise O2. For series cells, n(product) = n(e^-) / (electrons per product in half-equation).

For addition polymers, redraw any monomer as C(R1)(R2)=C(R3)(R4) first -> change = to - and extend bonds outside brackets. For ester R1-COO-R2, split the single C-O bond: R1-COOH (acid) + HO-R2 (alcohol).

Qualitative Anion Law: ALWAYS test Carbonate (HNO3 -> CO2 effervescence) -> Sulfate (Ba(NO3)2 -> white BaSO4) -> Halide (AgNO3 -> white AgCl / cream AgBr / yellow AgI) using NITRIC acid only.

For state-at-T tables, write T_target directly between the MP and BP columns: Liquid strictly requires MP < T_target < BP (watch negative signs: -70 is between -100 and -40!). In diffusion tubes, lighter gas (smaller M_r) travels further.

Group 1 vs Group 17 opposite rules: Down Group 1 -> Reactivity UP, Melting Point DOWN. Down Group 17 -> Reactivity DOWN, Boiling Point UP. Extraction threshold: above Carbon (Al, Mg, Na) = Molten Electrolysis; below Carbon (Fe, Zn, Cu) = Carbon Reduction.

Work in millimoles (mmol = c [mol dm⁻³] × V [cm³]) throughout! Proton-balance law: (# of H+ in acid) × c_acid × V_acid = (# of OH- in base) × c_base × V_base. Always multiply aliquot moles by (V_flask / V_pipette) at the end.

For hydrated salts M·xH2O heated from m_hydrated to m_anhydrous: m_water = m_hydrated - m_anhydrous, and x = (m_water / 18) ÷ (m_anhydrous / M_r(anhydrous)). Watch for 1 : 1.33 -> 3 : 4 (Fe3O4) and 1 : 1.5 -> 2 : 3 (Fe2O3).

Scan statements for 'bacteria + mitochondria/nucleus' or 'root hair + chloroplasts'—eliminating that single false statement typically kills 3 of 5 options in 5 seconds. For I = A*M, convert mm -> µm (×1000) first.

For a cube of side L, SA:V = 6/L (halving side length doubles SA:V). On an osmosis % mass change graph, the x-intercept (0% change) is the exact isotonic intracellular solute concentration.

Track (Chromosomes, DNA mass) as a pair: Somatic G1 = (2n, 2c) -> After S-phase / Metaphase = (2n, 4c) -> Mitosis daughter = (2n, 2c); Meiosis I daughter = (n, 2c); Meiosis II gamete = (n, 1c).

Pedigree 10-second rules: (1) Two unaffected parents -> affected child = RECESSIVE (both parents are obligate carriers). (2) Unaffected father + affected daughter = AUTOSOMAL (cannot be X-linked recessive). (3) Unaffected sibling of aa child has P(Aa) = 2/3.

Spot non-standard ratios immediately: 2:1 live offspring ratio = homozygous lethal allele (1/4 die in utero); 1:2:1 phenotypic ratio = codominance/incomplete dominance (e.g. I^A I^B or coat colour); in multi-cross tables, identify the homozygous recessive test-cross row first.

In double-stranded DNA, %A + %G = 50% ALWAYS (so if %A = 18%, %G = 50% - 18% = 32% in 2 seconds!). For coding length: N amino acids = 3N template bases = 6N total double-stranded bases.

Memorise the 3 recombinant rules: (1) Same restriction enzyme cuts both donor & plasmid -> complementary sticky ends (H-bonds form spontaneously). (2) DNA ligase seals the sugar-phosphate phosphodiester backbone. (3) n cuts in a circular plasmid = n fragments.

Respiration gas check: Mammals (Anaerobic) = 0 O2 in, 0 CO2 out (Lactate only). Yeast (Anaerobic) = 0 O2 in, CO2 out (+ Ethanol). Low temp = low KE (reversible); High temp = active site denatured (irreversible).

Valve Rule: Valves are passive one-way doors that open when upstream P > downstream P and snap shut at the exact graph intersection where downstream P > upstream P. Inhalation = Contract -> Flatten -> Volume UP -> Pressure DOWN.

ADH Rule: More ADH = More Collecting Duct Permeability = More Water Reabsorbed into Blood = Less, More Concentrated Urine. In a healthy nephron: Protein in filtrate = 0; Glucose in urine = 0.

Nitrogen bacteria 3-way check: N2 -> NH4+ (Nitrogen-fixing, root nodules/lightning); NH4+ -> NO2- -> NO3- (Nitrifying, aerobic); NO3- -> N2 (Denitrifying, anaerobic waterlogged soil). Eutrophication O2 drop is caused by AEROBIC DECOMPOSERS.

Potometer 2-step check: (1) Halve capillary diameter to radius r = d/2 before computing V = pi*r²*h; (2) Divide by time in minutes/hours. For lamp distance d1 -> d2, multiply light intensity by (d1 / d2)².

Instant Lamarckian eliminator: any statement saying an organism 'mutates in order to survive' or 'antibiotics cause mutations' is 100% FALSE. Mutation is always random and precedes the selection pressure.

Two classic physiology traps to eliminate immediately: (1) Blood vessels NEVER move closer to the skin; (2) Electrical impulses NEVER cross a synapse (neurotransmitters diffuse). Reflex arc order: Stimulus -> Receptor -> Sensory -> Relay (CNS) -> Motor -> Effector.

Quadrat scaling: Total Population = (Mean per quadrat) × (Total Area / Quadrat Area)—for a 0.5 m × 0.5 m quadrat, Area = 0.25 m² so multiply per-quadrat mean by 4 to get density per m²! Lincoln index: N = (n1 × n2) / m2.