RBSE SS Mathematics key terms

A set of ordered pairs linking elements of one set to elements of another; the set of all first elements is the domain and the set of all second elements is the range.

A relation in which every element of the domain is paired with exactly one element of the range, so no two ordered pairs share the same first coordinate.

The complete set of input values for which a relation or function is defined; for a real function it is the set of real numbers giving a real output.

The set of all output values a function actually produces from its domain, distinct from the codomain which may contain unused elements.

A function where distinct inputs always give distinct outputs, also called injective; equivalently, f(a) = f(b) forces a = b.

A function whose range equals its codomain, so every element of the codomain is the image of at least one domain element; also called surjective.

A function that is both one-one and onto; only such functions possess an inverse function defined on the whole codomain.

A function formed by applying one function to the output of another, written (g∘f)(x) = g(f(x)); its domain must satisfy both functions' restrictions.

A function that has an inverse, which happens exactly when it is bijective; the inverse reverses each ordered pair, so f⁻¹(y) = x whenever f(x) = y.

A restricted range chosen for an inverse trigonometric function so it becomes one-one and onto, giving a single agreed output for each input.

The inverse of a trigonometric function, defined on a restricted domain with a principal value range, for example sin⁻¹x with range from minus pi by two to pi by two.

A rectangular arrangement of numbers in rows and columns, enclosed in brackets; an m by n matrix has m rows and n columns and order m×n.

The pair stating how many rows and columns a matrix has, written rows first then columns; it fixes which matrix operations are possible with another matrix.

The matrix obtained by interchanging rows and columns, denoted Aᵀ; transposing twice returns the original matrix, and (AB)ᵀ = BᵀAᵀ.

A square matrix equal to its own transpose, so entry aᵢⱼ equals aⱼᵢ for all positions; its entries mirror across the main diagonal.

A square matrix whose transpose is its negative, so aᵢⱼ = −aⱼᵢ; every diagonal entry must be zero, and adding such a matrix to its transpose gives the zero matrix.

A single number computed from a square matrix that tells whether the matrix is invertible and how it scales areas or volumes; it is zero for singular matrices.

A square matrix whose determinant equals zero, so it has no inverse and its rows or columns are linearly dependent.

The transpose of the cofactor matrix of a square matrix, used in the formula A⁻¹ = adj(A)/|A| when the determinant is nonzero.

A method solving a square system of linear equations by writing each unknown as a ratio of two determinants, valid only when the coefficient determinant is nonzero.

A system of linear equations that has at least one solution; it has a unique solution when the coefficient determinant is nonzero and infinitely many when the system is dependent.

A property of a function at a point where the left limit, right limit and function value all exist and are equal; graphically the curve has no break there.

A property of a function at a point where the derivative exists as a finite limit of the difference quotient; differentiability at a point implies continuity there, but not conversely.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of the difference quotient; geometrically it gives the slope of the tangent.

An interior point of a function's domain where the derivative is zero or fails to exist; candidates for local maxima, local minima or points of inflection.

A point where the concavity of a curve changes sign, typically where the second derivative is zero or undefined while the curve remains continuous.

The way a curve bends: concave up when the second derivative is positive and concave down when it is negative, determined on intervals of the domain.

The family of all antiderivatives of a function, written with an arbitrary constant of integration because differentiating any member returns the original function.

The limit of a sum over a closed interval, interpreted as the signed area between a curve and the axis; it is a number found using the fundamental theorem of calculus.

A technique based on the product rule, written as the integral of u dv equals uv minus the integral of v du, used when an integrand is a product of unlike functions.

Rewriting a rational integrand as a sum of simpler fractions with linear or quadratic denominators, so each part can be integrated by standard formulas.

The region bounded by a curve, the horizontal axis and two vertical lines, computed as a definite integral; portions below the axis count as negative unless absolute values are used.

An equation involving an unknown function and one or more of its derivatives, classified by order and degree and solved by finding a function satisfying it.

The order of the highest derivative appearing in the equation; a first-order equation involves only the first derivative of the unknown function.

The power of the highest-order derivative after the equation is cleared of radicals and fractions in derivatives; defined only for polynomial equations in derivatives.

The solution of a differential equation containing as many arbitrary constants as the order of the equation, representing a whole family of curves.

A solution obtained from the general solution by assigning specific values to the arbitrary constants, usually fixed by given initial conditions.

A multiplier, for a first-order linear equation dy/dx + Py = Q it equals e to the integral of P dx, that makes the left side an exact derivative.

A quantity having both magnitude and direction, represented by a directed line segment and written in component form with unit vectors along the axes.

A vector of magnitude one, often used to give direction; any nonzero vector divided by its own magnitude becomes a unit vector in the same direction.

The scalar obtained from two vectors as the product of their magnitudes and the cosine of the angle between them; it is zero when the vectors are perpendicular.

A vector perpendicular to two given vectors with magnitude equal to the product of their magnitudes and the sine of the angle between them; it vanishes for parallel vectors.

The dot product of one vector with the cross product of two others; its absolute value gives the volume of the parallelepiped formed by the three vectors.

The cosines of the angles a line makes with the positive coordinate axes, usually denoted l, m and n; their squares sum to one.

An optimisation problem seeking the maximum or minimum of a linear objective function subject to linear inequality constraints and non-negativity restrictions.

The set of all points satisfying every constraint of a linear programming problem simultaneously; it is usually a convex polygon in the first quadrant.

The linear expression whose value is to be maximised or minimised in a linear programming problem, evaluated at the corner points of the feasible region.

A procedure solving a linear programming problem by evaluating the objective function at each vertex of the feasible region, since the optimum occurs at a vertex when it exists.

The probability of an event given that another event has already occurred, written P(A|B) and equal to P(A intersection B) divided by P(B) when P(B) is nonzero.

Two events whose occurrence does not affect each other's probability; equivalently, the probability of both equals the product of their separate probabilities.

Two events that cannot occur together, so their intersection is empty and the probability of either equals the sum of their probabilities.

A rule reversing a conditional probability, expressing P(A|B) in terms of P(B|A), the prior probability of A and the total probability of B.

A rule giving the probability of an event as the sum of its conditional probabilities over a partition of the sample space into mutually exclusive exhaustive events.

A real-valued function defined on the sample space of an experiment, assigning a number to each outcome so probabilities can be described numerically.

Repeated independent trials of an experiment with exactly two outcomes, success or failure, where the probability of success stays constant in every trial.

The probability distribution of the number of successes in a fixed number of independent Bernoulli trials, with formula nCr p^r q^(n−r) for r successes.

A organised collection of a student's mathematical work, such as solved problems, notes and reflections, assessed for understanding, effort and consistent record-keeping.

An extended independent investigation applying syllabus mathematics to a real situation, presented with aim, method, data, analysis and conclusion, and assessed on process as well as result.