Lebanese University Faculty of Engineering Entrance Examination key terms

The quantity b² − 4ac computed from a quadratic ax² + bx + c. Its sign tells whether the equation has two distinct real roots, one repeated real root, or a pair of complex conjugate roots.

The expression x = (−b ± √(b² − 4ac)) / (2a) giving all roots of ax² + bx + c = 0. It follows from completing the square and depends on the discriminant under the radical.

Relations linking coefficients of a polynomial to its roots: for ax² + bx + c the sum of roots is −b/a and the product is c/a. They allow root questions without solving.

A number of the form a + bi where i² = −1 and a, b are real. The real part a and imaginary part b are added and multiplied by ordinary rules with i² replaced by −1.

For z = a + bi, the modulus is |z| = √(a² + b²), its distance from the origin, and the argument is the angle θ with tan θ = b/a, giving the polar form z = |z|(cos θ + i sin θ).

States that [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) for integer n. It converts powers and roots of complex numbers into simple scaling and angle rotation.

The exponent to which a fixed base must be raised to produce a given number: log_b x = y means bʸ = x. Logarithms turn products into sums and are defined only for positive arguments with base b > 0, b ≠ 1.

A function of the form f(x) = aˣ with constant base a > 0, a ≠ 1. It is always positive, strictly monotonic, and its inverse is the logarithm with the same base.

The set of all input values for which the function is defined. Typical restrictions come from division by zero, even roots of negative numbers, and logarithms of non-positive numbers.

The function f⁻¹ that reverses f, so f⁻¹(f(x)) = x. It exists only when f is one-to-one on its domain, and its graph is the reflection of f's graph in the line y = x.

The set of all output values actually produced by a function as its input runs over the domain. For a quadratic with positive leading coefficient the range is bounded below by the vertex value.

The turning point of the graph of a quadratic, located at x = −b/(2a). It is the minimum when a > 0 and the maximum when a < 0, and it fixes the axis of symmetry.

A sequence in which each term differs from the previous one by a constant difference d. The nth term is u₁ + (n − 1)d and the sum of n terms is n(u₁ + uₙ)/2.

A sequence in which each term is the previous term multiplied by a fixed ratio q. The nth term is u₁qⁿ⁻¹, and the sum converges only when |q| < 1.

The value L that the terms of a sequence approach as the index grows without bound. A sequence converges when such a finite L exists and diverges otherwise.

A function whose graph has no breaks, jumps, or holes on its interval: the limit at each interior point equals the function value there. Continuity on a closed interval guarantees extreme values.

The instantaneous rate of change of a function, defined as the limit of the difference quotient [f(x + h) − f(x)]/h as h tends to zero. Geometrically it is the slope of the tangent line.

The straight line touching a curve at a point with slope equal to the derivative there. Its equation at x = a is y = f(a) + f′(a)(x − a).

An interior point of a function's domain where the derivative is zero or does not exist. Local extrema can occur only at critical points or at endpoints of the interval.

A point where a function reaches a local maximum or minimum compared with nearby values. A sign change of the derivative around a critical point identifies which kind occurs.

States that a function continuous on a closed interval takes every value between its endpoint values. It is the standard tool for proving an equation has a root in a given interval.

If a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then some c in (a, b) satisfies f′(c) = 0. It underlies the mean value theorem.

The limit of Riemann sums of a function over an interval [a, b], interpreted as the signed area between the graph and the x-axis. It is computed using an antiderivative evaluated at the endpoints.

A function F whose derivative equals a given function f, so F′ = f. Any two antiderivatives differ by a constant, and one is needed to evaluate a definite integral.

The region enclosed between two graphs from x = a to x = b, computed as the integral of the upper function minus the lower function. Intersection points usually set the limits of integration.

The line perpendicular to the tangent at a point of a curve. Its slope is the negative reciprocal of the derivative at that point, that is −1/f′(a).

For vectors u and v, the scalar u·v = |u||v|cos θ, also computed as the sum of products of components. It is zero exactly when the vectors are perpendicular.

A vector operation producing a vector perpendicular to both inputs, with magnitude |u||v|sin θ and direction given by the right-hand rule. It vanishes when the vectors are parallel.

The coordinates of a vector along chosen axes, so v = x i + y j (+ z k). They allow addition, scalar multiplication, and magnitude √(x² + y² + z²) to be handled algebraically.

In any triangle, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. It solves triangles when a side and its opposite angle are known.

States c² = a² + b² − 2ab cos C for a triangle with sides a, b, c and included angle C. It generalizes Pythagoras and finds a side from two sides and their included angle.

Ways to compute a triangle's area: half base times height, (1/2)ab sin C using two sides and the included angle, or √(s(s − a)(s − b)(s − c)) with semiperimeter s.

The net force on a body equals its mass times acceleration, ΣF = ma, measured in a fixed inertial frame. It links force analysis to the kinematics of motion.

A sketch showing one body isolated with every external force drawn as an arrow from the body: weight, normal force, friction, tension, and applied forces. It is the starting point for applying Newton's second law.

Equations relating displacement, initial and final velocity, acceleration, and time for constant acceleration, such as v = v₀ + at and v² = v₀² + 2aΔx. They apply only while acceleration stays constant.

Motion of a body under gravity alone after launch, with constant horizontal velocity and constant vertical acceleration g. The two components are treated independently and combined through time.

The force opposing the start of sliding between surfaces at rest, with maximum value μsN. It adjusts up to that limit to keep relative motion from beginning.

For a conductor at constant temperature, the current through it is proportional to the potential difference across it, U = RI, where R is the resistance in ohms.

In series, resistors share the same current and their resistances add. In parallel, they share the same voltage and the reciprocals of resistance add to give the reciprocal of the equivalent resistance.

Two rules for circuits: the junction rule states currents into a node equal currents out, and the loop rule states the sum of potential differences around any closed loop is zero.

The rate at which a component converts electrical energy, given by P = UI = RI² = U²/R. It determines heat dissipated in a resistor and energy drawn from a source.

The force on a charge q moving with velocity v in electric field E and magnetic field B: F = q(E + v × B). The magnetic part is perpendicular to both v and B and does no work.

The measure of magnetic field passing through a surface, Φ = B·A·cos θ, in webers. Changing this flux through a circuit is what induces an electromotive force.

The production of an electromotive force in a circuit by a changing magnetic flux through it, described by Faraday's law. Lenz's law fixes the direction so the induced current opposes the change.

The amount of substance containing 6.022 × 10²³ elementary entities. It converts between mass and number of particles through molar mass and Avogadro's number.

The mass of one mole of a substance in grams per mole, numerically equal to its relative formula mass. It is the bridge between a weighed mass and an amount in moles.

The reactant consumed first in a chemical reaction, because the others are present in excess. It fixes the maximum amount of product that can form and controls the theoretical yield.

The actual mass of product obtained divided by the theoretical mass predicted from the limiting reactant, multiplied by 100. It measures how efficient a reaction was in practice.

The concentration of a solution expressed as moles of solute per litre of solution, in mol/L. It links volume of solution to amount of substance in titration and dilution calculations.

The stage of a titration where the acid and base have reacted in exact stoichiometric proportions. It is detected with an indicator or pH meter and is not always at neutral pH.

A measure of acidity defined as pH = −log[H₃O⁺] for dilute aqueous solutions. A change of one pH unit corresponds to a tenfold change in hydrogen ion concentration.

A bookkeeping charge assigned to an atom in a compound by agreed rules, such as oxygen usually −2 and hydrogen usually +1. Changes in it identify oxidation and reduction.

A reaction in which electrons are transferred, so one species is oxidized (oxidation number rises) and another is reduced (oxidation number falls). The two changes always occur together.

The procedure of adjusting coefficients so atoms and total charge balance, using half-reactions for oxidation and reduction. Electrons lost must equal electrons gained before the halves are added.

Regular variations across the periodic table: atomic radius decreases and electronegativity and ionization energy increase from left to right, while radius increases down a group.

The tendency of an atom in a bond to attract shared electrons. Large differences between bonded atoms produce polar or ionic bonds; small differences give nonpolar covalent bonds.

The energy required to remove one electron from a gaseous atom or ion. It increases across a period and decreases down a group as the outer electron lies farther from the nucleus.

A specific group of atoms within an organic molecule that determines its characteristic reactions, such as hydroxyl in alcohols or carboxyl in acids. It guides naming and reactivity.

The systematic rules for naming organic compounds: choose the longest chain containing the principal functional group, number to give it the lowest locant, and list substituents alphabetically.

The existence of compounds with the same molecular formula but different structures or spatial arrangements. Structural isomers differ in connectivity, while stereoisomers differ only in three-dimensional layout.