Tamil Nadu SSLC Mathematics key terms

The set of all ordered pairs 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. It links elements of A with elements of B, and a relation on a single set is a subset of A × A.

A relation from set A to set B in which every element of A is paired with exactly one element of B. No input is left out, and no input has two different outputs.

A function in which different elements of the domain always give different images, so f(x₁) = f(x₂) forces x₁ = x₂. Also called an injective function or an injection.

The new function (f ∘ g)(x) = f(g(x)), obtained by applying g first and then f to the result. It is defined only where g(x) lies in the domain of f.

For any positive integers a and b, there exist unique whole numbers q and r with a = bq + r and 0 ≤ r < b. It underlies the Euclidean algorithm for finding the HCF.

Every composite number can be written as a product of primes in exactly one way, apart from the order of the factors. This unique prime factorisation gives HCF and LCM easily.

Two integers a and b are congruent modulo n when n divides a − b, written a ≡ b (mod n). They leave the same remainder on division by n.

A sequence in which each term after the first is obtained by adding a fixed number, the common difference, to the previous term. Its nth term is a + (n − 1)d.

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio. Its nth term is ar^(n−1).

The arithmetic series a + (a + d) + ... evaluated as (n/2)[2a + (n − 1)d], or as (n/2)(first term + last term). The formula applies only when the common difference is constant.

An expression of the form a₀ + a₁x + a₂x² + ... + aₙxⁿ where the coefficients are real numbers and the powers are whole numbers. Its degree is the highest power with a non-zero coefficient.

A value of x at which the polynomial takes the value zero. On the graph of y = p(x) these values are exactly the points where the curve meets the x-axis.

The polynomial of highest degree that divides both given polynomials without remainder. It is found by factorising each polynomial and multiplying the common factors with the smallest exponent.

The quantity b² − 4ac for the quadratic ax² + bx + c = 0. Its sign tells whether the roots are real and distinct, real and equal, or not real.

A rectangular arrangement of numbers in rows and columns, written inside brackets. Its order is given as rows × columns, and the entry aᵢⱼ lies in row i and column j.

A single number associated with a square matrix. For a 2 × 2 matrix it is ad − bc, and for a 3 × 3 matrix it is expanded using minors and cofactors.

An equation of the form ax² + bx + c = 0 with a ≠ 0. Its roots are the values of x that satisfy it, found by factorisation or by the quadratic formula.

Triangles whose corresponding angles are equal and whose corresponding sides are in the same ratio. They have the same shape but not necessarily the same size.

If a line is drawn parallel to one side of a triangle and cuts the other two sides, it divides those two sides in the same ratio. The converse is also true.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse decides whether a given triangle is right-angled.

A line that touches the circle at exactly one point, called the point of contact. It is perpendicular to the radius drawn to that point.

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from applying the Pythagoras theorem to the horizontal and vertical differences.

The coordinates of the point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m : n are ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

The tangent of the angle a line makes with the positive direction of the x-axis; also the ratio of vertical change to horizontal change. It equals (y₂ − y₁)/(x₂ − x₁) for two points.

The acute angle θ between lines of slopes m₁ and m₂ satisfies tan θ = |(m₁ − m₂)/(1 + m₁m₂)|. The angle is 90° when m₁m₂ = −1.

An equation involving trigonometric ratios that is true for every value of the angle for which it is defined. The standard example is sin²θ + cos²θ = 1.

Two angles whose sum is 90°. For such a pair, the sine of one angle equals the cosine of the other, tangent equals cotangent, and secant equals cosecant.

When looking at an object above the horizontal level of the eye, the angle between the horizontal line and the line of sight to the object.

The part of a cone left after cutting it by a plane parallel to its base and removing the smaller cone. Its volume uses the radii of both circular ends and the height.

The length along the sloping surface of a cone from the rim of the base to the apex. For a right circular cone it equals √(r² + h²).

The area of only the curved or lateral surface of a solid, excluding its flat faces. For a cylinder it is 2πrh and for a cone it is πrl.

The positive square root of the variance. It measures how far the observations spread around the mean, expressed in the same units as the data.

The mean of the squared deviations of the observations from their mean. It is the square of the standard deviation and can never be negative.

Two or more events that cannot happen at the same time, so their intersection is empty. For such events P(A ∪ B) = P(A) + P(B).

The set of all possible outcomes of a random experiment. Each outcome is equally likely in the classical definition of probability, which uses counts taken from this set.