Lebanese Baccalaureate - General Sciences (GS) cheat sheet

Quadratic formula: roots of ax²+bx+c=0 are x = (−b ± √(b²−4ac))/(2a); discriminant Δ=b²−4ac gives two real roots if Δ>0, one if Δ=0, none real if Δ<0.

Derivative rules: (xⁿ)′=nxⁿ⁻¹, (uv)′=u′v+uv′, (u/v)′=(u′v−uv′)/v², and (f∘g)′=f′(g(x))·g′(x).

ln and exp: ln(ab)=ln a+ln b, ln(a/b)=ln a−ln b, ln(aⁿ)=n ln a, and e^(ln x)=x for x>0.

Arithmetic progression: uₙ=u₁+(n−1)r, sum Sₙ=n(u₁+uₙ)/2; geometric: uₙ=u₁qⁿ⁻¹, Sₙ=u₁(1−qⁿ)/(1−q) for q≠1.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B); independence means P(A∩B)=P(A)P(B).

Limit definition of derivative: f′(a)=lim_{h→0}[f(a+h)−f(a)]/h; f is increasing where f′>0 and decreasing where f′<0.

Integration: ∫xⁿdx=xⁿ⁺¹/(n+1)+C for n≠−1; ∫dx/x=ln|x|+C; definite integral equals area under curve between bounds.

Complex numbers: i²=−1, modulus |a+bi|=√(a²+b²), conjugate of a+bi is a−bi, and z·z̄=|z|².

Vectors: dot product u·v=|u||v|cos θ; u⊥v iff u·v=0; cross-product magnitude |u×v|=|u||v|sin θ gives parallelogram area.

Newton's second law: ΣF=ma; weight W=mg with g≈9.8 m/s² near Earth's surface; friction f=μN.

Kinematics with constant acceleration: v=u+at, s=ut+½at², v²=u²+2as.

Work-energy: W=F·d·cos θ; kinetic energy KE=½mv²; gravitational potential energy near Earth PE=mgh; mechanical energy conserved without friction.

Momentum p=mv is conserved in collisions; impulse J=FΔt=Δp; elastic collisions conserve kinetic energy, inelastic ones do not.

Electricity: Ohm's law U=RI; power P=UI=RI²=U²/R; resistors in series add, in parallel 1/R=1/R₁+1/R₂.

Waves: v=fλ; for a string or standing wave, harmonics relate to length by λₙ=2L/n with fixed ends.

Radioactive decay: N=N₀e^(−λt), half-life T=ln2/λ; activity A=λN measured in becquerels.

Circular motion: centripetal acceleration a=v²/r; gravitational force F=Gm₁m₂/r² with G≈6.67×10⁻¹¹ N·m²/kg².

Mole relations: n=m/M, N=n·N_A with N_A≈6.02×10²³ mol⁻¹, and molar gas volume ≈22.4 L at STP.

Concentration: C=n/V (mol/L); dilution C₁V₁=C₂V₂; pH=−log[H₃O⁺] and pOH=−log[OH⁻], with pH+pOH=14 at 25 °C.

Acid-base titration at equivalence: moles of H₃O⁺ donated equal moles of OH⁻ accepted, using the stoichiometric coefficients.

Oxidation is loss of electrons (increase in oxidation number); reduction is gain; oxidation numbers in a neutral compound sum to zero.

Equilibrium constant K=[products]^coefficients/[reactants]^coefficients; Le Chatelier's principle predicts shift when concentration, pressure, or temperature changes.

Organic families: alkanes CₙH₂ₙ₊₂ (single bonds), alkenes CₙH₂ₙ (C=C), alkynes CₙH₂ₙ₋₂ (C≡C), alcohols R–OH.

Electrolysis: at the cathode reduction occurs, at the anode oxidation; quantity of charge Q=I·t and 1 mol electrons carries ≈96500 C.

Enthalpy: ΔH<0 for exothermic reactions; reaction rate increases with temperature, concentration, surface area, or a catalyst.

DNA structure: double helix of two antiparallel strands; adenine pairs with thymine (2 H-bonds), guanine with cytosine (3 H-bonds).

Central dogma: DNA is transcribed to mRNA in the nucleus, then translated to protein at ribosomes using triplet codons.

Mitosis produces two genetically identical diploid cells; meiosis produces four haploid cells and generates genetic variation by crossing-over.

Mendel's laws: segregation of alleles and independent assortment; a monohybrid cross Aa×Aa gives a 3:1 phenotypic ratio.

Cellular respiration: C₆H₁₂O₆+6O₂→6CO₂+6H₂O+energy (ATP); photosynthesis is the reverse, using light energy and chlorophyll.

Enzymes are protein catalysts with an active site; activity depends on pH and temperature, and denaturation destroys the active site.

Human blood circulation: double circulation with systemic and pulmonary circuits; the left ventricle pumps oxygenated blood to the body.

Nerve impulse: resting potential is negative inside; depolarization involves Na⁺ entry and repolarization involves K⁺ exit.

Integrated problems: convert all data to SI units before substituting into any physics or chemistry formula.

Graph reading rule: slope of a linear graph equals the ratio of the axis quantities; area under the curve often equals a physical quantity (e.g., v–t area = distance).

Significant figures: the result carries the same number of significant figures as the least precise given measurement.

Cross-domain rule: energy released in a chemical reaction per mole can be compared with mechanical work using ΔH=n·ΔH_molar and W=Fd.

Experimental design: identify the independent variable, dependent variable, and controlled variables before interpreting results.

Error analysis: relative error = |measured−accepted|/accepted ×100%; uncertainty propagates when quantities are multiplied or divided.

Biological measurements often use scales: magnification = image size / actual size; keep units consistent when computing.

Data interpretation: a correlation between two variables does not establish causation; a control group is needed to attribute an effect.

A scientific paper structure: title, abstract, introduction, methods, results, discussion, conclusion, references.

Hypothesis must be testable and falsifiable; it predicts a relationship between an independent and a dependent variable.

Validity concerns whether a method measures what it claims; reliability concerns whether repeated trials give consistent results.

Sampling: a representative sample reduces bias; larger sample size generally reduces random error.

Ethics in research: informed consent, confidentiality, and avoidance of harm are required when human or animal subjects are involved.

Peer review checks originality, methodology, and clarity before publication; it does not guarantee that results are correct.

Referencing avoids plagiarism; every borrowed idea or data point must be cited in the text and listed in the bibliography.

Interpreting a graph in a paper: check axis labels and units, note whether the relationship is linear or nonlinear, and identify outliers.

Scientific method order: observation, question, hypothesis, experiment, analysis, conclusion, communication.

A controlled experiment changes one variable at a time while keeping all others constant to isolate cause and effect.

A theory is a well-tested explanation supported by repeated evidence; a law describes a consistent relationship, often mathematically.

Models simplify reality; their predictions are valid only within the assumptions used to build them.

Falsifiability: a claim that cannot be tested and potentially disproved is not scientific.

Correlation versus causation: observational data can show association but cannot prove a causal mechanism without controlled testing.

Measurement precision depends on the instrument's smallest division; accuracy is closeness to the true value.

Reproducibility requires reporting methods and data in enough detail that another researcher can repeat the experiment.