Karnataka II PUC Mathematics (KSEAB) key terms

The set of all ordered pairs (a, b) formed by taking a from set A and b from set B; written A × B. If A has m elements and B has n, the product has mn elements.

Any subset of A × B. A relation from A to B is therefore just a collection of ordered pairs, and its domain and range are the sets of first and second coordinates appearing in those pairs.

A relation on a set that is reflexive, symmetric and transitive. It splits the set into disjoint equivalence classes whose union is the whole set.

A relation R on set A in which (a, a) belongs to R for every element a of A. The identity relation is the smallest reflexive relation on a set.

A relation R on A such that whenever (a, b) lies in R, the pair (b, a) also lies in R. Equality of numbers is symmetric; 'less than' is not.

A relation R on A where (a, b) and (b, c) in R always forces (a, c) into R. Divisibility and 'is a subset of' are standard transitive examples.

A function f: A → B where distinct inputs always give distinct outputs; f(x₁) = f(x₂) implies x₁ = x₂. Its graph meets every horizontal line at most once.

A function f: A → B whose range equals its codomain, so every element of B is the image of at least one element of A.

A function that is both one-one and onto. Only bijections possess an inverse function defined on the whole codomain.

For a bijection f: A → B, the function f⁻¹: B → A that reverses it, so f⁻¹(f(x)) = x and f(f⁻¹(y)) = y. Its graph is the reflection of f's graph in the line y = x.

The function gof defined by (gof)(x) = g(f(x)), applied only where f(x) lies in the domain of g. Composition is generally not commutative.

The restricted range chosen for an inverse trigonometric function so that it becomes one-one. For sin⁻¹ it is [−π/2, π/2]; for cos⁻¹ it is [0, π].

The inverse sine function, defined for x in [−1, 1] with values in [−π/2, π/2]. It answers the question: which angle in that interval has sine equal to x?

The inverse cosine function, defined for x in [−1, 1] with principal values in [0, π]. It is strictly decreasing on its domain.

The inverse tangent function, defined for all real x with principal values in (−π/2, π/2). Useful identities include tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)] when xy < 1.

The size of a matrix stated as rows × columns, for example 3 × 2. Two matrices can be added only when their orders are identical.

The matrix Aᵀ obtained by interchanging rows and columns of A, so the element in row i, column j moves to row j, column i. Note (AB)ᵀ = BᵀAᵀ.

A square matrix equal to its own transpose, so aᵢⱼ = aⱼᵢ for all i, j. Every symmetric matrix can be written as the sum of a symmetric and a skew-symmetric part.

A square matrix with Aᵀ = −A, which forces every diagonal entry to be zero. Its order must make the sign pattern consistent across the diagonal.

The product AB is defined only when the column count of A equals the row count of B; entry (i, j) of AB is the dot product of row i of A with column j of B. It is not commutative.

A scalar assigned to a square matrix that measures whether the matrix is invertible; it is zero exactly when the matrix is singular. For 2 × 2 it equals ad − bc.

The transpose of the cofactor matrix of a square matrix A, written adj A. It satisfies A(adj A) = (adj A)A = |A| I.

For a square matrix A with nonzero determinant, A⁻¹ = (adj A)/|A|, satisfying AA⁻¹ = A⁻¹A = I. No inverse exists when the determinant is zero.

A square matrix whose determinant is zero, so it has no multiplicative inverse. Its rows (or columns) are linearly dependent.

f is continuous at x = a when f(a) is defined, lim(x→a) f(x) exists, and the two are equal. All three conditions must hold together.

f is differentiable at a when the derivative f′(a) = lim(h→0)[f(a+h) − f(a)]/h exists. Differentiability at a point implies continuity there, but not conversely.

The rule for differentiating a composite: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). It underlies nearly every implicit and parametric derivative.

A technique in which you take logarithms of both sides before differentiating, turning products and powers into sums. It is the standard route for functions of the form [f(x)]^g(x).

If f 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 guarantees a horizontal tangent somewhere inside.

If f is continuous on [a, b] and differentiable on (a, b), then some c in (a, b) gives f′(c) = [f(b) − f(a)]/(b − a). The tangent at c is parallel to the chord joining the endpoints.

An interior point of the domain where f′(x) = 0 or f′ does not exist. Local maxima and minima can occur only at critical points or endpoints.

A test that classifies a critical point by the sign change of f′ around it: positive to negative gives a local maximum, negative to positive gives a local minimum, no change gives neither.

At a critical point c, f″(c) < 0 means a local maximum and f″(c) > 0 means a local minimum. The test is inconclusive when f″(c) = 0.

The family of all antiderivatives of a function, written ∫f(x)dx = F(x) + C where F′ = f. The arbitrary constant C must always be included.

The number ∫ₐᵇ f(x)dx, defined as the limit of Riemann sums and equal to F(b) − F(a) for any antiderivative F. It represents signed area under the curve.

It links the two branches of calculus: if f is continuous on [a, b] and F is any antiderivative, then ∫ₐᵇ f(x)dx = F(b) − F(a). It also gives d/dx ∫ₐˣ f(t)dt = f(x).

The rule ∫u dv = uv − ∫v du, used for products of unlike functions. Choose u by the ILATE order: inverse, logarithmic, algebraic, trigonometric, exponential.

Writing a proper rational function as a sum of simpler fractions with linear or quadratic denominators, so each piece can be integrated separately.

The order of the highest derivative appearing in the equation. dy/dx = x is first order; d²y/dx² + y = 0 is second order.

The power of the highest-order derivative once the equation is written as a polynomial in its derivatives. It is defined only when such a polynomial form exists.

The solution of a differential equation containing as many arbitrary constants as the order of the equation. A particular solution is obtained by fixing those constants from given conditions.

For the linear equation dy/dx + Py = Q, the multiplier e^∫P dx that makes the left side exact, so the solution follows as y·(integrating factor) = ∫Q·(integrating factor)dx.

The scalar a·b = |a||b|cos θ, also computed as the sum of products of corresponding components. It is zero exactly when the vectors are perpendicular.

The vector a × b with magnitude |a||b|sin θ and direction perpendicular to both, given by a determinant with i, j, k in the first row. It is zero for parallel vectors.

The number [a b c] = a·(b × c), equal to the determinant of the three vectors' components. Its absolute value gives the volume of the parallelepiped they span.

The cosines of the angles a line makes with the positive x, y and z axes, usually written l, m, n. They satisfy l² + m² + n² = 1.

Any three numbers proportional to the direction cosines of a line. Unlike direction cosines, they need not satisfy a unit-sum condition.

For lines r = a₁ + λb₁ and r = a₂ + μb₂, the distance is |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|. It is zero when the lines intersect or are parallel.

The set of points satisfying all constraints of a linear programming problem simultaneously, usually a convex polygon. Optimal solutions occur at its vertices.

The linear expression Z = ax + by that a linear programming problem seeks to maximise or minimise, subject to the stated constraints.

A procedure that evaluates the objective function at every vertex of the feasible region; the largest and smallest values are the maximum and minimum. It works because the optimum lies at a vertex.

The probability of A given that B has occurred, P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. It restricts the sample space to B.

Events A and B with P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Knowing one occurred gives no information about the other.

A rule that reverses conditional probability: P(Aᵢ|E) = P(Aᵢ)P(E|Aᵢ) / Σ P(Aⱼ)P(E|Aⱼ). It updates prior probabilities in the light of observed evidence E.

Repeated independent trials with exactly two outcomes, success or failure, and constant success probability p. The number of successes in n trials follows a binomial distribution.

The distribution of the number of successes in n Bernoulli trials: P(X = r) = C(n, r) pʳ (1−p)^(n−r). Its mean is np and variance np(1−p).