Cuba 12th Grade Math Final Exam flashcards

30 cards. Read the term and the definition here. The deck at the top of this page is optional.

Vector director

Non-zero vector parallel to a line; together with one point of the line it determines the line completely in vector, parametric and continuous forms.

Vector normal

Non-zero vector perpendicular to a plane; its components are exactly the coefficients of x, y and z in the general equation of the plane.

Producto escalar

Operation on two vectors giving the number |u||v|cos θ; zero result means the vectors are perpendicular, and it yields the angle between them.

Producto vectorial

Operation on two vectors in space giving a third vector perpendicular to both, with modulus |u||v|sin θ; used to build normals and to compute areas of parallelograms.

Producto mixto

Scalar u·(v×w), equal to the determinant of the three vectors as rows; it gives the signed volume of the parallelepiped and vanishes when they are coplanar.

Vectores coplanarios

Three or more vectors lying parallel to a single plane; three of them are coplanar exactly when their mixed product is zero.

Ecuaciones paramétricas de la recta

Form x = a₁ + t·v₁, y = a₂ + t·v₂, z = a₃ + t·v₃, where (a₁,a₂,a₃) is a point, v the direction vector and t the real parameter.

Ecuación general del plano

Equation Ax + By + Cz + D = 0, where (A,B,C) is a normal vector; substituting a point of the plane fixes the constant D.

Posición relativa de rectas

Classification of two lines as coincident, parallel, intersecting or skew, decided by comparing their direction vectors and checking whether a vector joining points lies in their span.

Rectas que se cruzan

Two lines in space that are neither parallel nor intersecting, so they lie in different parallel planes and share no point.

Ángulo entre recta y plano

For a nonzero line direction v and plane normal n, the non-obtuse angle alpha satisfies sin(alpha)=abs(v·n)/(|v||n|). It is complementary to the acute or right angle between the line direction and the normal.

Distancia de un punto a un plano

Length of the perpendicular segment from the point to the plane, computed as |Ax₀+By₀+Cz₀+D| divided by the modulus of the normal vector (A,B,C).

Combinaciones

Selections of k elements from n without regard to order, counted by C(n,k) = n!/(k!(n−k)!); used when only the group matters.

Permutaciones

Ordered arrangements of all or part of a set of elements; the number of orderings of n distinct items is n!, and repeated items require division by their factorials.

Probabilidad condicionada

Probability of A given that B has occurred, P(A|B) = P(A∩B)/P(B), defined when B has non-zero probability.

Independencia de sucesos

Two events are independent when P(A∩B) = P(A)·P(B), so knowing one occurred does not change the probability of the other.

Teorema de Bayes

Formula reversing a conditional probability, P(A|B) = P(B|A)P(A)/P(B), with the denominator expanded over a partition of the sample space.

Diagrama de árbol

Graph whose branches represent successive stages of an experiment with probabilities on the edges; path products give intersections and sums give totals.

Distribución binomial

Model for the number of successes in n independent trials with constant success probability p; probabilities come from C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np.

Distribución normal

Continuous symmetric bell-shaped distribution fixed by mean μ and standard deviation σ; probabilities are areas under the curve found by standardising to N(0,1).

Tipificación

Change of variable z = (x − μ)/σ that converts any normal variable into the standard normal N(0,1), allowing use of a single table.

Aproximación normal de la binomial

Use of a normal with mean np and variance np(1−p) to estimate binomial probabilities when n is large and p is not extreme, applying a continuity correction.

Intervalo de confianza

Range of plausible values for an unknown population parameter, built from a sample statistic plus or minus a margin set by the confidence level.

Error estándar de la media

Standard deviation of the sample mean, equal to σ/√n, or to s/√n when the population standard deviation is unknown; it shrinks as the sample grows.

Teorema central del límite

For independent identically distributed observations with finite variance, the standardized sample mean approaches a normal distribution as sample size increases. The approximation quality depends on population shape and sample size; it is not guaranteed for every arbitrary population.

Contraste de hipótesis

Procedure comparing a null hypothesis about a parameter with sample data, deciding rejection when the standardised statistic falls in a critical region.

Mean

The arithmetic average of a data set, found by summing all values and dividing by the count; sensitive to extreme values or outliers.

Median

The middle value of an ordered data set, or the average of the two middle values when the count is even; resistant to outliers compared with the mean.

Standard Deviation

A measure of the typical distance of data values from the mean, equal to the square root of the variance; larger values indicate more spread.

Normal Distribution

A symmetric, bell-shaped probability distribution defined by its mean and standard deviation; roughly 68% of values fall within one standard deviation of the mean.