ENP Alger Access Contest cheat sheet

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a); discriminant Δ = b² − 4ac gives two real roots if Δ > 0, one double root if Δ = 0, no real roots if Δ < 0.

Logarithm rules: log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x − log_a y, log_a(xⁿ) = n log_a x, and log_a b = ln b / ln a for any valid base change.

Derivative shortcuts: (xⁿ)′ = n xⁿ⁻¹, (eˣ)′ = eˣ, (ln x)′ = 1/x, (sin x)′ = cos x, (cos x)′ = −sin x; product rule (uv)′ = u′v + uv′ and quotient rule (u/v)′ = (u′v − uv′)/v².

Sum of arithmetic progression: S_n = n(a₁ + a_n)/2 = n(2a₁ + (n−1)d)/2; sum of geometric progression: S_n = a₁(1 − rⁿ)/(1 − r) for r ≠ 1.

Combinatorics: number of arrangements of n distinct objects is n!; choosing k from n without order is C(n,k) = n!/(k!(n−k)!); with order it is P(n,k) = n!/(n−k)!.

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); if independent, P(A ∩ B) = P(A)P(B); conditional probability P(A|B) = P(A ∩ B)/P(B).

Trigonometric identity and special values: sin²θ + cos²θ = 1; sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = √2/2, sin 60° = √3/2.

Vectors in the plane: dot product u·v = |u||v|cos θ = u₁v₁ + u₂v₂; perpendicular iff u·v = 0; magnitude |u| = √(u₁² + u₂²).

Limits and continuity rule: a function is continuous at a point if the two-sided limit exists and equals the function value there; lim(x→0) sin x / x = 1 and lim(x→∞)(1 + 1/x)ˣ = e.

Kinematics with constant acceleration: v = v₀ + at, x = v₀t + ½at², v² = v₀² + 2ax; these apply only when acceleration is constant.

Newton's second law: ΣF = ma; weight is W = mg with g ≈ 9.8 m/s² near Earth's surface; friction magnitude f = μN where N is the normal force.

Work–energy theorem: net work equals change in kinetic energy, W_net = ΔK = ½mv² − ½mv₀²; gravitational potential energy near Earth is U = mgh.

Conservation of mechanical energy when only conservative forces act: ½mv₁² + mgh₁ = ½mv₂² + mgh₂.

Momentum: p = mv; in an isolated system total momentum is conserved; perfectly inelastic collision combines masses with v = (m₁v₁ + m₂v₂)/(m₁ + m₂).

Ohm's law and power in circuits: V = IR, P = VI = I²R = V²/R; resistors in series add, in parallel 1/R_eq = Σ1/R_i.

Waves: speed v = fλ; for a wave on a string v = √(T/μ) with tension T and linear density μ; sound intensity level in decibels β = 10 log(I/I₀) with I₀ = 10⁻¹² W/m².

Ideal gas law: PV = nRT with R ≈ 8.31 J/(mol·K); at constant temperature Boyle's law gives P₁V₁ = P₂V₂.

Radioactive decay: N = N₀e^(−λt); half-life T½ = ln 2 / λ ≈ 0.693/λ.

Mole relations: n = m/M (mass over molar mass), n = N/N_A with N_A ≈ 6.022 × 10²³ mol⁻¹, and at STP one mole of ideal gas occupies about 22.4 L.

Molarity: c = n/V with V in litres; dilution obeys c₁V₁ = c₂V₂ because moles of solute are conserved.

pH definition: pH = −log[H₃O⁺]; for a strong monoprotic acid [H₃O⁺] equals acid concentration; pH + pOH = 14 at 25 °C.

Balancing redox: oxidation number rises on oxidation and falls on reduction; total electrons lost must equal total electrons gained.

Acid–base neutralisation stoichiometry: moles of H⁺ equal moles of OH⁻ at equivalence; for a diprotic acid one mole supplies two moles of H⁺.

Le Chatelier's principle: increasing concentration of a reactant shifts equilibrium toward products; increasing pressure shifts toward the side with fewer gas moles; a catalyst changes rate, not position.

Bonding: ionic bonds form between metal and non-metal with large electronegativity difference; covalent bonds share electrons; a polar bond arises from unequal sharing.

Gas stoichiometry uses the ideal gas law PV = nRT; volume ratios of gases at the same T and P equal mole ratios from the balanced equation.

Organic functional groups: alcohols −OH, carboxylic acids −COOH, aldehydes −CHO, ketones C=O between carbons, esters −COO−; alkanes follow CₙH₂ₙ₊₂.

Cell theory: all living organisms are made of cells, the cell is the basic unit of life, and new cells arise from existing cells by division.

Photosynthesis overall: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂; light reactions occur in thylakoid membranes and the Calvin cycle in the stroma.

Cellular respiration: glycolysis in the cytoplasm yields 2 ATP and pyruvate; the Krebs cycle and oxidative phosphorylation in mitochondria yield most ATP.

DNA structure: two antiparallel strands with sugar–phosphate backbones; adenine pairs with thymine by two hydrogen bonds, guanine with cytosine by three.

Mendelian genetics: a monohybrid cross of two heterozygotes gives a 3:1 phenotype ratio and 1:2:1 genotype ratio; a test cross uses a homozygous recessive partner.

Protein synthesis: transcription copies DNA into mRNA in the nucleus; translation at ribosomes reads codons three bases at a time, each codon specifying one amino acid.

Enzyme kinetics: enzymes lower activation energy; rate rises with substrate concentration until saturation; each enzyme has an optimum temperature and pH; competitive inhibitors raise apparent K_m.

Homeostasis: negative feedback reverses a deviation (e.g. insulin lowers blood glucose); positive feedback amplifies a change (e.g. oxytocin in labour).

Ecosystem energy flow: about 10 % of energy transfers between trophic levels; decomposers return nutrients to the soil; producers form the base of every food web.

Controlled experiment rule: change one independent variable at a time, hold all others constant, and measure the dependent variable with a stated unit.

Correlation versus causation: an observed association does not establish cause; a causal claim needs controlled manipulation or a well-justified natural experiment.

Hypothesis quality: a scientific hypothesis must be testable and falsifiable; a statement that no possible observation could contradict is not scientific.

Significant figures: multiplication and division keep the smallest number of significant figures in the inputs; addition and subtraction keep the smallest number of decimal places.

Unit analysis: a valid equation must be dimensionally consistent on both sides; converting units by multiplying by ratios equal to one preserves value.

Graph interpretation: slope of a linear graph is Δy/Δx with units of y per x; the intercept is the value of y when x is zero; a curved trend signals a non-linear relation.

Sampling and bias: a random sample supports generalisation to the population; a convenience sample may be biased, and a larger sample reduces random error but not systematic error.

Percent change and error: percent change = (new − old)/old × 100; percent error = |measured − accepted|/accepted × 100.

Logical inference: from 'if P then Q' and P, conclude Q (valid modus ponens); from 'if P then Q' and Q, concluding P is the fallacy of affirming the consequent.