PSEB Class 12 Maths key terms

A relation on a set that is reflexive, symmetric and transitive. It partitions the set into disjoint classes whose members are related to one another.

A function where different inputs always give different outputs; if f(a)=f(b), then a=b. Such a function never repeats an output value.

A function whose codomain equals its range: every element of the codomain is the image of at least one element of the domain.

A function that is both injective and surjective, so each output corresponds to exactly one input. Only bijections have inverse functions defined on the whole codomain.

The combined function (f∘g)(x)=f(g(x)), formed when the output of g becomes the input of f. Its domain contains inputs x for which g(x) lies in the domain of f.

A chosen restricted domain on which an inverse trigonometric function is one-to-one, giving a single agreed output. For example, sin⁻¹ has principal range from negative pi by two to positive pi by two.

The domain is the set of all allowed input values of a function; the range is the set of all output values it actually produces.

A square matrix equal to its transpose, so a_ij equals a_ji for all row and column indices. Its entries are mirror images across the main diagonal.

A square matrix whose transpose is its negative, so a_ij equals negative a_ji and every diagonal entry is zero. Only square matrices can have this property.

A scalar assigned to a square matrix that measures whether the matrix is invertible. It is zero exactly when the rows or columns are linearly dependent.

The transpose of the cofactor matrix of a square matrix. Multiplying a matrix by its adjoint gives the determinant times the identity matrix.

For a square matrix A, the matrix A⁻¹ satisfying AA⁻¹=A⁻¹A=I. It exists only when det(A) is nonzero, and then equals adj(A) divided by det(A).

The signed minor obtained by deleting the row and column of an entry and multiplying its minor determinant by (-1)^(i+j). Cofactors build determinants and adjoints.

A function f is continuous at x=a when f(a) is defined, the limit as x approaches a exists, and that limit equals f(a).

A function is differentiable at a point when its derivative there exists as a finite limit. Differentiability at a point implies continuity there, but continuity does not imply differentiability.

A rule for differentiating composite functions: the derivative of f(g(x)) is f'(g(x)) times g'(x). It is applied from the outside inward.

A method for finding dy/dx when y is not isolated, by differentiating both sides of an equation with respect to x and treating y as a function of 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.

If f is continuous on [a,b] and differentiable on (a,b), then some c in (a,b) satisfies f'(c) = [f(b)-f(a)]/(b-a).

An interior point of a function's domain where the derivative is zero or does not exist. Critical points are candidates for local extrema.

A value f(c) that is at least as large as nearby function values: f(c) ≥ f(x) for all x in some open interval containing c.

The family of all antiderivatives of a function, written ∫f(x)dx = F(x)+C, where F'(x)=f(x) and C is an arbitrary constant.

The limit of Riemann sums over an interval [a,b], representing signed area under a curve. It is a number, evaluated using antiderivatives via the fundamental theorem.

A technique from the product rule: ∫u dv = uv - ∫v du. It is useful when an integrand is a product of unlike functions such as x and e^x.

It links differentiation and integration: if F is an antiderivative of continuous f, then the integral from a to b of f(x)dx equals F(b)-F(a).

The definite integral of a nonnegative function over an interval, giving the region between its graph and the x-axis. For curves crossing the axis, signed areas must be handled separately.

An equation involving an unknown function and one or more of its derivatives. Its solution is a function, often containing arbitrary constants unless initial conditions are given.

The order is the highest derivative present; the degree is the power of that highest derivative after the equation is cleared of radicals and fractions in derivatives.

A function μ(x) that, when multiplied through a linear first-order differential equation, makes the left side an exact derivative, allowing direct integration.

A vector drawn from the origin to a point, whose components equal the point's coordinates. It specifies the location of the point relative to the origin.

The scalar a·b = |a||b|cosθ, also equal to the sum of products of corresponding components. It measures alignment and tests perpendicularity.

A vector a×b perpendicular to both a and b, with magnitude |a||b|sinθ and direction given by the right-hand rule. It gives area and torque.

The length of the common perpendicular between two non-intersecting, non-parallel lines. It equals |(a₂-a₁)·(b₁×b₂)|/|b₁×b₂| for position vectors a₁,a₂ and direction vectors b₁,b₂.

The set of all points satisfying every constraint of a linear programming problem, including nonnegativity. The optimum, if it exists, occurs at a vertex of this region.

The linear expression to be maximised or minimised in a linear programming problem, such as profit or cost. Its value is evaluated over the feasible region.

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

Events A and B are independent when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0. Occurrence of one does not change the other's probability.

A formula that reverses a conditional probability: P(A|B) = P(B|A)P(A)/P(B), provided P(B)>0. It updates a prior probability for A using observed evidence B.

A function assigning a numerical value to each outcome of a random experiment. It may be discrete or continuous, with an associated probability distribution.

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