CHSE Odisha Elective Mathematics key terms

The set of all ordered pairs whose first element comes from one set and second element from another. Written A × B, it contains every pair (a, b) with a in A and b in B.

Any subset of the Cartesian product A × B. It links elements of one set to elements of another and may be given by a rule, a list of ordered pairs, or an arrow diagram.

A relation in which every element of the domain is paired with exactly one element of the codomain. Two different inputs may share an output, but one input cannot produce two outputs.

The domain is the full set of inputs a function accepts; the range is the set of values it actually produces. The codomain may be larger than the range.

A function in which distinct inputs always give distinct outputs, so f(a) = f(b) forces a = b. On a graph, no horizontal line meets the curve more than once.

A relation that is reflexive, symmetric and transitive. It splits the underlying set into disjoint classes whose members are all related to one another, such as congruence of triangles.

The single agreed output chosen from the many possible ones of an inverse trigonometric function. It lies in a fixed interval, for example minus pi by two to plus pi by two for inverse sine.

A function has an inverse only when it is one-one and onto on the stated interval. Trigonometric functions are therefore restricted, such as cosine to zero to pi, before inversion is defined.

A single number computed from a square matrix that records whether the matrix can be inverted. A zero determinant means the matrix is singular; a nonzero one means its inverse exists.

The minor of an entry is the determinant left after deleting its row and column. The cofactor is that minor multiplied by the sign factor (−1)^(i+j); cofactors build both the adjoint and the expansion of a determinant.

The transpose of the matrix formed from the cofactors. Multiplying a matrix by its adjoint gives the determinant times the identity, which produces the inverse formula A inverse equals adjoint A divided by determinant A.

A square matrix equal to the negative of its transpose, so every diagonal entry is zero. Its determinant vanishes for odd order, and the definition is used to test whether a given matrix has this symmetry.

The complex number with the same real part but the opposite imaginary part. Multiplying a number by its conjugate gives a non-negative real value, which is how division and modulus are handled.

The modulus is the distance of a complex number from the origin in the Argand plane; the argument is the angle its position vector makes with the positive real axis, taken in a standard range.

For the quadratic ax² + bx + c = 0, the value b² − 4ac. Its sign decides whether the roots are real and distinct, real and equal, or a pair of complex conjugates.

An arrangement of objects in a definite order. The number of permutations of r objects chosen from n distinct objects is n factorial divided by (n − r) factorial, and order is always counted.

A selection of objects in which order does not matter. The count is n factorial divided by r factorial times (n − r) factorial, always smaller than the matching permutation count by the factor r factorial.

The number written C(n, r) that multiplies each term in the expansion of (a + b)ⁿ. It equals the corresponding combination count and is symmetric, since C(n, r) equals C(n, n − r).

A sequence with a constant difference between consecutive terms. Its nth term is a + (n − 1)d, and the sum of n terms is n/2 times the sum of the first and last terms.

A sequence in which each term is a fixed multiple of the previous one. Its nth term is a·r^(n−1) and, when the modulus of r is less than one, the infinite sum equals a/(1 − r).

The value the outputs of a function approach as the input approaches a chosen point, whether or not the function is defined there. Left and right limits must agree for the limit to exist.

A function is continuous at a point when its limit there exists, equals the value of the function at that point, and the function is defined there. Graphically the curve has no break or hole.

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

The derivative defined as the limit, as h tends to zero, of f(x + h) minus f(x) all divided by h. This definition underlies every standard derivative formula and is often examined directly.

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

A point where the function value is at least as large as the values at nearby points. At such an interior point of a differentiable function the first derivative is zero, and a sign change confirms it.

The largest value a continuous function takes on a closed interval. It occurs either at a critical point inside the interval or at one of the two endpoints, so all must be checked.

The family of all antiderivatives of a function, written with an arbitrary constant. Because constants disappear on differentiation, every integrable function has infinitely many antiderivatives differing only by constants.

The number obtained by integrating a function between two limits. It equals the signed area between the curve and the horizontal axis, with regions below the axis counted as negative.

A rule derived from the product rule, written integral of u dv equals uv minus integral of v du. Choosing u as the factor that simplifies when differentiated makes otherwise difficult integrals manageable.

The order is the highest derivative appearing in the equation; the degree is the power of that highest derivative once the equation is written as a polynomial in its derivatives.

A multiplying factor that turns a linear first-order differential equation into an exact one. For dy/dx + Py = Q it equals e raised to the integral of P dx, allowing the equation to be integrated directly.

The dot product of one vector with the cross product of two others. Its absolute value equals the volume of the parallelepiped they span, and it is zero exactly when the three vectors are coplanar.

The cosines of the angles that a line makes with the positive coordinate axes. Their squares add up to one, and they are the components of the unit vector along that line.

The length of the common perpendicular between two lines that neither meet nor are parallel. It is found using the scalar triple product of their direction vectors and a vector joining a point on each line.

The set of all points that satisfy every constraint of a linear programming problem simultaneously. If it is bounded and non-empty, an optimum lies at a corner; if it is empty, no solution exists.

Evaluating the objective function at each vertex of the feasible region and comparing the results. For a bounded region the largest or smallest of these values is guaranteed to be the optimum.

The probability of an event given that another event has already occurred, written P(A|B) equals P(A and B) divided by P(B). It effectively rescales the sample space to the known event.

The probability distribution of the number of successes in n independent trials, each having the same success probability p. Its mean is np and its variance is np(1 − p).