Balearic Islands PAU Mathematics II key terms

Matrix obtained by turning rows into columns and columns into rows; entry (i,j) of the transpose equals entry (j,i) of the original matrix.

Square matrix that equals its own transpose, so entries mirrored across the main diagonal are identical; A = Aᵗ.

Matrix B such that A·B = B·A = I, the identity. It exists only for square matrices with non-zero determinant, and then B = adj(A)ᵗ/|A|.

Scalar assigned to a square matrix that is zero exactly when the matrix is singular; computed by cofactor expansion or by row operations that track how the value changes.

Determinant of the submatrix left after deleting one row and one column of a square matrix; the associated cofactor multiplies it by (−1)^(i+j).

Transpose of the cofactor matrix of A; dividing it by the determinant of A produces the inverse matrix when that determinant is non-zero.

Maximum number of linearly independent rows, equal to the maximum number of independent columns; found by reducing the matrix to echelon form and counting non-zero rows.

Rule classifying a linear system by comparing the rank of the coefficient matrix with the rank of the augmented matrix and with the number of unknowns.

Linear system with at least one solution and exactly one; occurs when the ranks of coefficient and augmented matrices both equal the number of unknowns.

Linear system with infinitely many solutions; the common rank of coefficient and augmented matrices is smaller than the number of unknowns.

Linear system with no solution at all, detected when the rank of the augmented matrix exceeds the rank of the coefficient matrix.

Letter appearing in the coefficients or constants of a system; the discussion studies for which values the system is compatible and how many solutions it has.

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.

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.

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.

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.

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.

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

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.

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

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.

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

Angle between a line and its orthogonal projection on a plane; obtained from the sine of the angle between the line direction and the plane normal.

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).

Value L that f(x) approaches as x approaches a point; it may exist even when the function is undefined or takes another value at that point.

Property of a function at a point where the limit exists, the function is defined there, and both values coincide; differentiability implies it but not conversely.

Expression such as 0/0, ∞/∞, ∞−∞ or 0·∞ whose limit cannot be read off directly and requires algebraic manipulation, L'Hôpital's rule or equivalent techniques.

Method for indeterminate quotients: replace the limit of f/g by the limit of the quotient of derivatives f'/g', valid when the original limit has form 0/0 or ∞/∞.

Limit of the difference quotient, giving the slope of the tangent line and the instantaneous rate of change of a function at a point.

Existence of the derivative at a point; it requires continuity there, and at a breakpoint the two one-sided derivatives must agree.

Line touching a curve at a point with slope equal to the derivative there; its equation is y − f(a) = f'(a)(x − a).

Search for maxima or minima of a function, usually by solving f'(x) = 0 and classifying the critical points with the sign of the second derivative or a sign table.

If a function is continuous on a closed interval and takes values of opposite sign at its ends, then it vanishes at some interior point; used to locate roots.

For a function continuous on [a,b] and differentiable on (a,b) with equal endpoint values, there is an interior point where the derivative is zero.

Function F whose derivative is the given function f; all primitives differ from F by an additive constant, forming the indefinite integral.

Limit of Riemann sums over an interval, equal to F(b) − F(a) for any primitive F; it gives the signed area between the curve and the x-axis.

Fundamental theorem statement used in practice: the definite integral from a to b equals a primitive evaluated at b minus the same primitive evaluated at a.

Technique based on ∫u dv = uv − ∫v du, useful when the integrand is a product such as a polynomial times an exponential or logarithm.

Region bounded by two graphs between two intersection abscissas, computed as the integral of the absolute difference of the functions.

Space generated by rotating a region about an axis; for rotation about OX the volume is π times the integral of f(x)² over the interval.

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.

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.

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

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

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.

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

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.

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).

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.

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.

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

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.

For large samples the distribution of the sample mean is approximately normal whatever the population shape, which justifies normal-based confidence intervals.

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