Lebanese University Faculty of Engineering Entrance Examination cheat sheet

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

Vieta's relations: for ax² + bx + c = 0, sum of roots = −b/a and product = c/a; for a cubic ax³ + bx² + cx + d = 0, sum = −b/a and product of all three = −d/a.

Logarithm rules: log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x − log_a y, log_a(x^n) = n log_a x, log_a b = ln b / ln a, and a^(log_a x) = x. Arguments must be positive.

Exponential equation a^x = a^y implies x = y only when a > 0 and a ≠ 1; if 0 < a < 1 the function is strictly decreasing, so inequalities reverse when taking logs.

Domain rules: a fraction requires nonzero denominator; an even root requires a nonnegative radicand; a logarithm requires a positive argument. Intersect all restrictions for composite functions.

Absolute value: |x| = x for x ≥ 0 and −x for x < 0; |xy| = |x||y|; the inequality |x| < k is equivalent to −k < x < k, while |x| > k means x < −k or x > k.

Arithmetic sequence: u_n = u_1 + (n − 1)r, sum S_n = n(u_1 + u_n)/2. Geometric sequence: u_n = u_1·q^(n−1), sum S_n = u_1(1 − q^n)/(1 − q) for q ≠ 1.

Asymptotes of a rational function: vertical where the simplified denominator vanishes; horizontal from the ratio of leading coefficients when numerator and denominator degrees are equal, y = 0 when the denominator degree is larger.

Pythagorean identity: sin²θ + cos²θ = 1, hence 1 + tan²θ = 1/cos²θ and 1 + cot²θ = 1/sin²θ, valid wherever the functions are defined.

Addition formulas: sin(a ± b) = sin a cos b ± cos a sin b; cos(a ± b) = cos a cos b ∓ sin a sin b; tan(a ± b) = (tan a ± tan b)/(1 ∓ tan a tan b).

Double angle: sin 2a = 2 sin a cos a; cos 2a = cos²a − sin²a = 2cos²a − 1 = 1 − 2sin²a; tan 2a = 2 tan a/(1 − tan²a).

Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. Cosine rule: a² = b² + c² − 2bc cos A.

Triangle area: S = ½·b·c·sin A, and Heron's formula S = √[p(p − a)(p − b)(p − c)] with semiperimeter p = (a + b + c)/2.

Circle: equation (x − h)² + (y − k)² = r² has centre (h, k) and radius r; the line y = mx + c is tangent when the discriminant of the substituted equation is zero.

Vectors: dot product u·v = |u||v|cos θ = x₁x₂ + y₁y₂; u ⊥ v exactly when u·v = 0; the angle satisfies cos θ = (u·v)/(|u||v|).

Solid volumes: cone and cylinder V = ⅓πr²h and πr²h; sphere V = (4/3)πr³ with surface area 4πr²; a plane section parallel to the base of a cone gives a smaller similar cone.

Derivative definition: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h; differentiability at a point implies continuity there, but continuity does not imply differentiability.

Standard derivatives: (x^n)′ = n·x^(n−1), (sin x)′ = cos x, (cos x)′ = −sin x, (e^x)′ = e^x, (ln x)′ = 1/x, (tan x)′ = 1/cos²x.

Product, quotient, chain rules: (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²; [f(g(x))]′ = f′(g(x))·g′(x).

Limits: lim_{x→0} (sin x)/x = 1, lim_{x→0} (1 − cos x)/x² = ½, lim_{x→0} (e^x − 1)/x = 1, lim_{x→0} ln(1 + x)/x = 1.

L'Hôpital's rule applies to indeterminate forms 0/0 or ∞/∞: lim f/g = lim f′/g′ provided the latter limit exists.

Monotonicity: f′ > 0 on an interval means f is strictly increasing there, f′ < 0 strictly decreasing; a sign change of f′ at a critical point with f′ = 0 gives a local extremum.

Integration by parts: ∫u dv = uv − ∫v du; substitution reverses the chain rule, and the definite integral changes limits accordingly.

Definite integral: ∫_a^b f(x)dx = F(b) − F(a) for any antiderivative F; the value equals the signed area between the curve and the x-axis.

Uniformly accelerated motion: v = v₀ + at, x = x₀ + v₀t + ½at², v² = v₀² + 2a(x − x₀); these hold only for constant acceleration.

Projectile motion: horizontal velocity is constant, vertical acceleration is −g; time of flight for launch speed v₀ at angle θ over level ground is 2v₀sin θ/g and range is v₀²sin 2θ/g.

Newton's second law: ΣF = ma in an inertial frame; weight is W = mg, and on an incline of angle α the components are mg sin α along the slope and mg cos α perpendicular to it.

Friction: static friction satisfies f ≤ μ_s·N, kinetic friction is f = μ_k·N acting opposite to relative sliding; N is the normal force, not necessarily mg.

Circular motion: centripetal acceleration a = v²/r = ω²r directed toward the centre; angular velocity ω = 2π/T with T the period.

Work and energy: W = F·d·cos θ; work–energy theorem W_net = ΔKE with KE = ½mv²; gravitational potential energy near the Earth is PE = mgh.

Momentum: p = mv; total momentum is conserved in collisions when net external impulse is negligible; elastic collisions also conserve kinetic energy, inelastic ones do not.

Static equilibrium of a rigid body requires ΣF = 0 and Στ = 0 about any point; torque τ = r·F·sin θ, and the moment of inertia of a point mass at distance r is mr².

Coulomb's law: F = k|q₁q₂|/r² with k ≈ 9 × 10⁹ N·m²/C²; the force is repulsive for like charges and attractive for unlike charges, directed along the joining line.

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

Kirchhoff's rules: the sum of currents entering a node equals the sum leaving; the algebraic sum of potential differences around a closed loop is zero.

Capacitor: Q = CU; energy stored E = ½CU² = Q²/(2C); capacitors in parallel add, in series reciprocals add.

Magnetic force on a moving charge F = qvB sin θ, direction from the right-hand rule; on a straight wire F = BIL sin θ. Flux through an area is Φ = BA cos θ.

Faraday–Lenz law: induced emf e = −dΦ/dt; the induced current opposes the change in flux, and a conductor of length L moving at speed v perpendicular to B develops emf BLv.

Wave relation v = fλ; for a string fixed at both ends the harmonic frequencies are f_n = n·v/(2L), and sound speed in air is about 340 m/s at room temperature.

Snell's law: n₁ sin i = n₂ sin r, with n = c/v; total internal reflection occurs only when light travels from a denser to a less dense medium beyond the critical angle sin i_c = n₂/n₁.

Amount of substance: n = m/M = N/N_A = V/V_m; at standard conditions V_m ≈ 22.4 L/mol, and N_A ≈ 6.02 × 10²³ mol⁻¹.

Molar concentration c = n/V in mol/L; dilution satisfies c₁V₁ = c₂V₂ because the amount of solute is unchanged.

pH = −log[H₃O⁺]; for a strong acid [H₃O⁺] equals its concentration, for a strong base pOH = −log[OH⁻] and pH + pOH = 14 at 25 °C.

Oxidation numbers: an element in its standard state is 0; oxygen is usually −2, hydrogen +1; the sum over a neutral molecule is 0 and over an ion equals its charge.

Redox: oxidation is loss of electrons, reduction is gain; the oxidising agent is reduced, and half-reactions are balanced by mass and charge before combining.

Periodic trends: atomic radius decreases across a period and increases down a group; ionisation energy and electronegativity increase across a period and decrease down a group.

Gas laws: PV = nRT with R ≈ 8.314 J/(mol·K); at constant T, P₁V₁ = P₂V₂, and at constant P, V₁/T₁ = V₂/T₂ with T in kelvin.

Electron configuration follows the Aufbau order, Pauli exclusion and Hund's rule; ions of main-group elements gain or lose electrons to reach the nearest noble-gas configuration.

Homologous series: alkanes CₙH₂ₙ₊₂, alkenes and cycloalkanes CₙH₂ₙ, alkynes CₙH₂ₙ₋₂; each carbon forms four bonds and each oxygen two.

IUPAC naming: the longest chain gives the parent name, numbering starts from the end nearest the first substituent, and the suffix fixes the principal functional group (‑ol, ‑al, ‑oic acid).

Isomerism: structural isomers differ in connectivity; geometric (cis–trans) isomerism requires restricted rotation, as in a C=C double bond with two different groups on each carbon.

Addition reactions of alkenes: H₂ with a catalyst, halogens giving dihalides, HX giving Markovnikov's major product, and water giving an alcohol.

Substitution: alkanes react with halogens under UV light by a radical mechanism; benzene undergoes electrophilic substitution rather than addition, preserving aromaticity.

Oxidation: primary alcohols give aldehydes then carboxylic acids, secondary alcohols give ketones, tertiary alcohols resist mild oxidation; aldehydes reduce Tollens' reagent, ketones do not.

Stoichiometry: balance the equation, convert masses to moles, apply the molar ratios, then convert back; the limiting reactant is the one giving the smaller amount of product.

Percentage yield = (actual mass / theoretical mass) × 100; combustion of a hydrocarbon CₙHₘ gives n CO₂ and m/2 H₂O, which allows formula determination from gas volumes.