Tamil Nadu HSE First Year Mathematics key terms

A well-defined collection of distinct objects, called its elements, treated as a single entity. Membership must be unambiguous, so one can always decide whether a given object belongs to it.

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

The set of all ordered pairs formed by taking the first element from set A and the second from set B, written A × B. It consists of all pairs (a, b) with a in A and b in B.

A subset of the Cartesian product A × B, so it consists of ordered pairs linking elements of A with elements of B. The set of first coordinates is called its domain.

A relation from set A to set B in which every element of A is paired with exactly one element of B. No input value may be associated with two different outputs.

A function in which different elements of the domain always produce different images, so f(a) = f(b) forces a = b. It is also called a one-one function.

The function formed by applying one function to the output of another, written (g ∘ f)(x) = g(f(x)). It is defined only where the inner output lies in the outer function's domain.

A function that reverses the action of another function, sending each image back to its original element. It exists only when the original function is both one-one and onto.

For the quadratic equation ax² + bx + c = 0, the quantity b² − 4ac. Its sign shows whether the roots are real and distinct, real and equal, or non-real.

When a polynomial f(x) is divided by x − a, the remainder equals f(a). This lets you find a remainder by substitution instead of carrying out long division.

A proper rational expression rewritten as a sum of simpler fractions whose denominators are the factors of the original denominator. This form makes integration and other algebraic work easier.

The angle subtended at the centre of a circle by an arc whose length equals the radius. One radian equals 180/π degrees, approximately 57.3 degrees.

The complete set of all angles satisfying a trigonometric equation, usually written with an integer n to cover every repetition of the solution. It contrasts with a single principal solution.

The product of all positive integers from 1 up to a given non-negative integer n, written n!. By convention the factorial of zero equals 1.

An arrangement of a selection of objects in a definite order, where changing the order gives a different permutation. The number of arrangements of r objects chosen from n is written nPr.

A selection of objects in which order does not matter, so choosing the same objects in a different order counts only once. The number of such selections is written nCr.

A proof technique for statements about natural numbers: prove the statement for n = 1, assume it true for n = k, then use that assumption to prove it for n = k + 1.

The number multiplying each term in the expansion of (a + b)^n, written nCr or C(n, r). It equals the number of ways of choosing r objects from n objects.

In the binomial expansion of (a + b)^n, the term containing a^(n−r) b^r, usually written T_(r+1). Giving r different values produces each successive term of the expansion.

A sequence in which each term after the first is obtained by adding a fixed constant, 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 constant, the common ratio. Its nth term is a r^(n−1).

The set of all points in a plane that satisfy a given geometric condition. The equation of a locus is the algebraic relation satisfied by the coordinates of exactly those points.

A measure of the steepness of a line, equal to the tangent of its angle with the positive x-axis. For two points it is the change in y divided by the change in x.

A number describing how much a conic section departs from being a circle. For a parabola it equals 1, while for an ellipse it is less than 1.

A rectangular arrangement of numbers or expressions in rows and columns, enclosed in brackets. Its order is written as rows × columns, and each entry is called an element.

A single number associated with a square matrix, computed from its entries by a fixed rule. A non-zero determinant means the matrix possesses a multiplicative inverse.

The signed minor of an element of a square matrix, obtained by multiplying its minor by (−1)^(i+j), where i and j are the row and column numbers of the element.

A square matrix whose determinant equals zero. It has no multiplicative inverse, and its rows or columns are linearly dependent on one another.

The scalar obtained from two vectors by multiplying their magnitudes and the cosine of the angle between them. It equals zero when the two vectors are perpendicular.

A vector perpendicular to two given vectors, whose magnitude equals the product of their magnitudes and the sine of the angle between them. It is zero for parallel vectors.

The value that the output of a function approaches as its input approaches a particular point. The limit may exist even when the function is not defined at that point.

A property of a function at a point where the function is defined, its limit exists, and the limit equals the function's value there. Intuitively, the graph has no break at that point.

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

A rule for differentiating composite functions: the derivative of f(g(x)) equals f′(g(x)) multiplied by g′(x). It is applied starting from the outer function and working inward.

The family of all antiderivatives of a function, written with an arbitrary constant of integration. Differentiating the result returns the original function.

The integral of a function between two limits, interpreted as a signed area and evaluated by subtracting antiderivative values at the endpoints. It gives a number, not a family of functions.

The set of all possible outcomes of a random experiment. Each outcome is an element of this set, and every event is a subset of it.

Two or more events that cannot occur at the same time, so their intersection is empty. Their combined probability is the sum of their individual probabilities.

The probability that an event occurs given that another event has already occurred, written P(A|B) and equal to P(A ∩ B) divided by P(B), where P(B) is not zero.