Kerala Plus Two Mathematics (DHSE) key terms

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

Any subset of the Cartesian product A × B, i.e. a rule linking some elements of A to elements of B. Its domain is the set of first components and its range the set of second components actually used.

A relation on a set that is reflexive, symmetric and transitive simultaneously. Such a relation splits the set into disjoint equivalence classes, and every element lies in exactly one class.

A function that is both one-one (distinct inputs give distinct outputs) and onto (every element of the codomain is hit). Only bijections possess an inverse function defined on the whole codomain.

The function (g ∘ f)(x) = g(f(x)), obtained by applying f first and then g. It is defined only where f(x) lies in the domain of g; composition is generally not commutative.

A restricted interval chosen for an inverse trigonometric function so that the original trigonometric function becomes one-one there. The inverse then returns exactly one value, called the principal value, lying in that interval.

The standard restrictions: sin⁻¹ and cos⁻¹ take input in [−1, 1] with ranges [−π/2, π/2] and [0, π]; tan⁻¹ takes all reals with range (−π/2, π/2). Cot⁻¹, sec⁻¹ and cosec⁻¹ have analogous fixed intervals.

Standard formulas such as sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, and 2tan⁻¹x = tan⁻¹(2x/(1 − x²)) for |x| < 1. They are used to simplify expressions and solve equations.

A square matrix A is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A. In a skew-symmetric matrix all diagonal entries are zero, and every square matrix splits into a symmetric plus a skew-symmetric part.

The matrix Aᵀ obtained by interchanging rows and columns of A, so the entry in row i, column j of Aᵀ equals the entry in row j, column i of A. It satisfies (AB)ᵀ = BᵀAᵀ.

A square matrix is singular when its determinant equals zero and non-singular when its determinant is non-zero. Only a non-singular matrix has an inverse, and it is used to test uniqueness of solutions of linear systems.

The transpose of the cofactor matrix of a square matrix A, written adj A. It satisfies A(adj A) = (adj A)A = |A|I, which gives the inverse formula A⁻¹ = (adj A)/|A| when |A| ≠ 0.

A scalar attached to a square matrix, computed by expanding along a row or column using cofactors. It measures whether the matrix is invertible and gives the area or volume scale factor of the associated transformation.

A method of solving a system of linear equations using determinants: each unknown equals the ratio of a determinant with that column replaced by the constants to the coefficient determinant, provided the latter is non-zero.

A system AX = B is consistent if at least one solution exists, which happens when the coefficient determinant is non-zero or when ranks of the coefficient and augmented matrices agree. Inconsistency means no solution exists.

A function f is continuous at x = a when the left-hand limit, the right-hand limit and f(a) all exist and are equal. Equivalently, lim over x approaching a of f(x) equals f(a).

A function is differentiable at a point when its derivative exists there, meaning the difference quotient has a single finite limit. Differentiability implies continuity, but continuity does not imply differentiability, as |x| at 0 shows.

The rule for differentiating a composite function: if y = f(u) and u = g(x), then dy/dx = f′(u) · g′(x). It is the standard tool for differentiating nested expressions such as sin(3x² + 1).

A technique for finding dy/dx when y is not isolated, as in an equation relating x and y. Each side is differentiated with respect to x, treating y as a function of x and applying the chain rule.

The value that f(x) approaches as x approaches a given point, possibly from one side only. Left and right limits must agree for the two-sided limit to exist; limits underlie both continuity and derivatives.

An interior point of a function's domain where the derivative is zero or fails to exist. Local maxima and minima can occur only at critical points or at endpoints of the interval considered.

A test classifying critical points by the sign change of f′ around them: positive to negative gives a local maximum, negative to positive gives a local minimum, and no sign change gives neither.

At a critical point c where f′(c) = 0, the sign of f″(c) decides the type: f″(c) < 0 gives a local maximum, f″(c) > 0 gives a local minimum, and f″(c) = 0 leaves the test inconclusive.

The largest and smallest values a function takes on a closed interval. They are found by evaluating the function at all critical points inside the interval and at both endpoints, then comparing.

The derivative dy/dx interpreted as how fast y changes per unit change in x. In applications, dy/dt measures the rate of change with respect to time, and related rates link two such quantities.

The family of antiderivatives of a function, written ∫f(x)dx = F(x) + C where F′ = f and C is an arbitrary constant. It is computed from standard formulas and by substitution or partial fractions.

The number ∫ from a to b of f(x)dx, defined as the limit of a Riemann sum and equal to the net signed area between the curve and the x-axis. It is evaluated using the fundamental theorem of calculus.

The rule ∫u dv = uv − ∫v du, obtained by reversing the product rule. It suits products such as x·eˣ or x·sin x, with u chosen to become simpler when differentiated.

The region bounded by y = f(x), the x-axis and vertical lines x = a and x = b, computed as the definite integral of f from a to b. Where the curve lies below the axis, the integral counts that area as negative.

The region enclosed by y = f(x) and y = g(x) between x = a and x = b, given by the integral of |f(x) − g(x)| over that interval. The upper curve minus the lower curve keeps the result positive.

The order is the highest derivative present; the degree is the power of the highest-order derivative after the equation is made polynomial in derivatives. Both are read off only after clearing radicals and fractions in derivatives.

The general solution of a differential equation contains as many arbitrary constants as its order; a particular solution is obtained by fixing those constants from given conditions, such as an initial value.

An equation that can be written with all y-terms on one side and all x-terms on the other, after which both sides are integrated. The constant of integration is added once, giving the general solution.

For a first-order linear equation dy/dx + Py = Q, the multiplier e^(∫P dx) that turns the left side into the derivative of a product, allowing direct integration. It is also used for homogeneous equations converted to linear form.

The dot product a·b = |a||b|cos θ is a scalar measuring alignment; the cross product a×b is a vector perpendicular to both, with magnitude |a||b|sin θ, giving the area of the parallelogram they span.

The quantity [a b c] = a·(b×c), a scalar whose absolute value equals the volume of the parallelepiped with edges a, b, c. It is zero exactly when the three vectors are coplanar.

The cosines of the angles a line makes with the positive x, y and z axes, usually denoted l, m, n. They satisfy l² + m² + n² = 1 and give the direction ratios when scaled by a common factor.

For lines with direction vectors b₁ and b₂ through points a₁ and a₂, the distance is |(a₂ − a₁)·(b₁×b₂)| / |b₁×b₂|. It measures the perpendicular gap between two non-intersecting, non-parallel lines.

The set of all points satisfying every constraint of a linear programming problem, including the non-negativity restrictions. It is a convex polygon in two variables, and the optimum of the objective occurs at one of its corner points.

The linear expression, such as Z = ax + by, that a linear programming problem seeks to maximise or minimise subject to given constraints. Its optimal value is attained at a vertex of the feasible region.

A procedure for solving a linear programming problem: graph the feasible region, find the coordinates of every corner, evaluate the objective function at each, and select the largest or smallest value as required.

The probability of event A given that B has occurred, written P(A|B) = P(A ∩ B)/P(B) with P(B) > 0. It restricts the sample space to B and underlies independence and Bayes' theorem.

Two events A and B are independent when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Knowing one occurred gives no information about the other; the multiplication rule then simplifies.

A rule reversing conditional probability: P(Eᵢ|A) equals P(Eᵢ)P(A|Eᵢ) divided by the sum of such terms over all mutually exclusive causes Eᵢ. It updates prior probabilities after observing A.

Repeated independent trials with exactly two outcomes and constant success probability p. The probability of r successes in n trials is nCr pʳ q^(n−r), where q = 1 − p; its mean is np and variance npq.

For a statement 'if p then q', the contrapositive is 'if not q then not p' and is logically equivalent to the original; the converse is 'if q then p' and is not equivalent. The inverse is 'if not p then not q'.

A proof technique for statements about natural numbers: verify the base case, assume the statement for n = k, then prove it for n = k + 1. Success in both steps establishes it for all natural numbers.

The average of the absolute deviations of observations from a central value such as the mean or median, computed as Σ|xᵢ − M|/n. It measures dispersion but is less used than variance because absolute values are awkward algebraically.

Variance is the mean of squared deviations from the mean, σ² = Σ(xᵢ − x̄)²/n; standard deviation is its positive square root. The standard deviation is expressed in the same units as the data.

The collection of all subsets of a given set, including the empty set and the set itself. A set with n elements has 2ⁿ subsets, so its power set has 2ⁿ elements.

If one task can be done in m ways and a second in n ways, the two together can be done in m × n ways when performed in succession, or m + n ways when exactly one of them is chosen.

An arrangement of objects in a definite order. The number of permutations of n distinct objects taken r at a time is nPr = n!/(n − r)!, since order matters.

A selection of objects where order does not matter. The number of ways to choose r objects from n distinct objects is nCr = n!/[r!(n − r)!], and nCr = nC(n − r).

The expansion (a + b)ⁿ = Σ nCr a^(n−r) bʳ for a non-negative integer n. The general term is nCr a^(n−r) bʳ, and the coefficients nCr form the corresponding row of Pascal's triangle.

The (r + 1)th term in the expansion of (a + b)ⁿ, written T(r+1) = nCr a^(n−r) bʳ. Setting the exponent of a variable to a required value in this expression locates a particular term.

A sequence in which each term differs from the previous one by a fixed constant d. The nth term is a + (n − 1)d and the sum of n terms is (n/2)[2a + (n − 1)d].

A sequence in which each term is obtained by multiplying the previous term by a fixed ratio r. The nth term is ar^(n−1) and the sum of n terms is a(rⁿ − 1)/(r − 1) for r ≠ 1.

For two positive numbers a and b, the arithmetic mean is (a + b)/2 and the geometric mean is √(ab). The arithmetic mean is at least the geometric mean, with equality only when a = b.