Karnataka II PUC Basic Mathematics (KSEAB) key terms

A well-defined collection of distinct objects treated as one entity. A rule decides membership, so every object either belongs to the collection or does not.

Set A is a subset of set B when every element of A also lies in B. Every set is a subset of itself, and the empty set is a subset of every set.

For sets A and B, the set of all ordered pairs (a, b) with a in A and b in B. If A has m elements and B has n, the product has mn elements.

A rule that assigns to each element of a set A exactly one element of a set B. A is the domain, B contains the range, and no input has two outputs.

A number computed from a square matrix that measures whether the matrix is invertible. For a 2x2 matrix with first row a, b and second row c, d it equals ad - bc.

The signed minor of an element of a square matrix: the determinant left after deleting its row and column, multiplied by (-1) raised to the sum of its row and column numbers.

The transpose of the matrix of cofactors of a square matrix. Multiplying a matrix by its adjoint gives the determinant times the identity matrix, which yields the inverse formula.

A square matrix whose determinant is zero, so it has no inverse. Its rows or columns are linearly dependent, and the system it represents may have no unique solution.

The product of all positive integers from 1 up to a given whole number n, written n!. By convention 0! = 1, and n! counts the arrangements of n distinct objects in a row.

An arrangement of a chosen number of objects from a group, where the order matters. The number of r-permutations of n distinct objects is n! divided by (n - r)!.

A selection of objects from a group where order does not matter. The number of r-combinations of n distinct objects is n! divided by r!(n - r)!.

In the binomial expansion of (a + b)^n, the term containing a^(n-r) b^r, written as nCr a^(n-r) b^r. Putting r = 0, 1, 2, ... generates each term in turn.

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 nonzero number, the common ratio. Its nth term is ar^(n-1).

For two positive numbers a and b, the number G with G squared equal to ab, so G equals the square root of ab. In a geometric progression each term is the geometric mean of its neighbours.

For a geometric progression with first term a and common ratio r where the size of r is less than 1, the total of all terms equals a divided by (1 - r). The series converges to this value.

The measure of steepness of a line, equal to the change in y divided by the change in x between two points. It is the tangent of the angle the line makes with the positive x-axis.

The equation y - y1 = m(x - x1) of a straight line with slope m passing through the point (x1, y1). It is used to write a line's equation from one point and its slope.

The perpendicular distance from a point (x1, y1) to the line ax + by + c = 0, given by the modulus of ax1 + by1 + c divided by the square root of (a squared + b squared).

The set of all points in a plane at a fixed distance, the radius, from a fixed point, the centre. Its standard equation is (x - h)^2 + (y - k)^2 = r^2.

The set of points in a plane equidistant from a fixed point, the focus, and a fixed line, the directrix. Standard forms include y^2 = 4ax, whose focus is (a, 0).

For a conic, the ratio of the distance of a point on it from the focus to its distance from the directrix. It equals 1 for a parabola, is below 1 for an ellipse and above 1 for a hyperbola.

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

A function is continuous at a point when its limit there exists, the function is defined there, and the limit equals the function's value. Informally, 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 the difference quotient as the change in the variable tends to zero. It gives the slope of the tangent.

A rule for differentiating a composite function: the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). It lets you differentiate nested expressions layer by layer.

A function is increasing on an interval when larger inputs give larger outputs there. If its derivative is positive throughout an interval, the function is increasing on that interval.

A point where a function's value is greater than at all nearby points. At such a point the derivative is zero or undefined, and the derivative changes from positive to negative.

The limit of a sum of areas of thin rectangles under a curve between two limits, written with lower and upper limits on the integral sign. It gives a signed area and is evaluated using antiderivatives.

The arithmetic average of a set of observations, found by adding all values and dividing by their number. It uses every observation but is pulled noticeably by extreme values.

The positive square root of the variance, expressing how far observations typically lie from the mean in the original units. A small value means the data cluster tightly around the mean.

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

A formula that reverses a conditional probability: P(A|B) equals P(B|A)P(A) divided by P(B). It updates a prior probability using evidence and is used with partitions of a sample space.

The linear expression in a linear programming problem whose value is to be maximised or minimised, such as profit or cost. It is evaluated at corner points of the feasible region.

The linear inequalities or equations that restrict the values of the decision variables in a linear programming problem, usually coming from limits on resources, time or demand. They define the feasible region.

The set of all points satisfying every constraint of a linear programming problem at once. It is usually a polygon, and the best value of the objective function occurs at a corner.

Interest calculated only on the original principal for the whole period, using I = PTR/100, where P is principal, T is time in years and R is the annual rate percent.

Interest calculated on the principal plus interest already added, so the amount grows as A = P(1 + R/100)^n. Interest is added at fixed intervals such as yearly or half-yearly.

A series of equal payments made at regular intervals, such as monthly or yearly. Its present or future value is found using geometric series formulas for the payments.

The simple interest on the face value of a bill for the period from the date of discounting to the legal due date. It is what the bank deducts, so the holder receives face value minus this discount.