Tamil Nadu HSE Second Year Mathematics key terms

The determinant of the smaller square matrix left after deleting the row and column containing a chosen element of a square matrix. It is a number used to build cofactors.

The signed minor of an element, obtained by multiplying its minor by plus or minus one depending on whether the sum of its row and column indices is even or odd.

The transpose of the matrix formed by replacing every element with its cofactor. Multiplying a matrix by its adjoint gives the determinant times the identity matrix.

For a square matrix with non-zero determinant, the matrix obtained by dividing the adjoint by that determinant; multiplying it with the original matrix gives the identity matrix.

A square matrix whose determinant equals zero. It has no inverse, and the linear system it represents either has no solution or infinitely many solutions.

The order of the largest square submatrix whose determinant is non-zero. It measures how many independent rows or columns the matrix contains.

A property of a set of linear equations meaning at least one solution exists. Comparing the ranks of the coefficient and augmented matrices decides consistency and uniqueness.

A triangular arrangement of a matrix in which the first non-zero entry of each row lies to the right of the one above, and rows of zeros sit at the bottom.

A systematic method of applying elementary row operations to reduce an augmented matrix to row echelon form, then solving the equations by back substitution.

The distance of a complex number from the origin and the angle its position vector makes with the positive real axis, together giving the polar form.

Writing a complex number as r times the sum of cosine theta and i sine theta, where r is the modulus and theta the argument.

A rule stating that the nth power of a complex number in polar form has modulus raised to n and argument multiplied by n; it also handles rational powers and roots.

The three solutions of z cubed equals one: one itself and two complex numbers whose sum is minus one and whose product is one.

The n distinct complex solutions of z to the power n equals a given complex number, spaced equally around a circle in the Argand plane.

The coordinate plane used to picture complex numbers, with the real part measured horizontally and the imaginary part vertically.

The complex number obtained by reversing the sign of the imaginary part. Its product with the original number equals the square of the modulus.

Expressions such as sum of roots, sum of products taken two at a time, and product of all roots, whose values follow directly from the coefficients of a polynomial.

For a polynomial written with leading coefficient one, the coefficient of each lower power equals a signed elementary symmetric function of the roots.

A polynomial equation whose coefficients read the same forwards and backwards, so that if a value is a root its reciprocal is also a root.

Producing a new equation whose roots are related to the old roots by a fixed rule, such as multiplying each root by a constant or taking reciprocals.

The sign of the discriminant of a quadratic decides whether its roots are real and distinct, real and equal, or a complex conjugate pair.

The restricted interval chosen for an inverse trigonometric function so that it becomes single valued, for example minus pi by two to pi by two for inverse sine.

The function returning the angle whose sine is a given number between minus one and one, with output taken in the principal interval from minus pi by two to pi by two.

Limiting the input set of a trigonometric function so that it is one-to-one, which is what allows its inverse to be defined as a genuine function.

The ratio of the distance of a point on a conic from the focus to its distance from the directrix; its value classifies the curve as ellipse, parabola or hyperbola.

The chord of a conic drawn through a focus perpendicular to the major or transverse axis; its length is a standard quantity in each conic's equation.

Two straight lines that the branches of a hyperbola approach ever more closely at great distances but never meet, found by replacing the constant term with zero.

A hyperbola whose asymptotes are perpendicular, occurring when the transverse and conjugate axes have equal lengths and the eccentricity equals root two.

Expressing the coordinates of any point on a conic in terms of a single parameter, such as a parameter t on a parabola or an angle on an ellipse.

The dot product of one vector with the cross product of two others; its absolute value gives the volume of the parallelepiped they span.

The cross product of two vectors, a vector perpendicular to both whose magnitude equals the area of the parallelogram they determine.

The condition that three vectors lie parallel to one plane, detected by their scalar triple product being zero.

The cosines of the angles a line makes with the three coordinate axes; their squares sum to one and they fix the line's direction.

The length of the common perpendicular joining two non-intersecting, non-parallel lines in space, computed using a scalar triple product of direction and position vectors.

A function is continuous at a point when its limit there exists, the value at the point exists, and the two are equal.

If a function has a derivative at a point then it must be continuous there; the converse fails, as sharp corners show.

A break in a function where the limit exists but differs from the function value or the value is missing, so redefining one point repairs it.

If a function is continuous on a closed interval, differentiable inside it, and equal at both ends, then its derivative vanishes somewhere inside.

For a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change over the interval.

A point where the first derivative is zero; the second derivative's sign then decides whether it is a local maximum, a local minimum, or neither.

A point where a curve changes the direction of its bending, typically where the second derivative changes sign.

A technique that rewrites the integral of a product as a product minus another integral, choosing one factor to differentiate and the other to integrate.

The value of a definite integral equals the signed area between the curve and the axis over the interval, with regions below the axis counted negative.

The volume generated when a plane region rotates about an axis, computed by integrating pi times the square of the radius over the interval.

A relation expressing an integral with a higher power or parameter in terms of a similar integral with a lower one, applied repeatedly until the base case is reached.

The order of the highest derivative appearing in the equation; it tells how many arbitrary constants the general solution will contain.

The power of the highest order derivative after the equation has been cleared of radicals and fractions in the derivatives.

The family of solutions containing as many arbitrary constants as the order of the equation, from which particular solutions are obtained by fixing values.

A multiplying factor, usually a function of one variable, that turns a non-exact first order linear differential equation into one whose left side is a derivative.

A table listing every possible combination of truth values of the component statements and the resulting value of a compound statement.

A compound statement is a tautology when it is true under every assignment of truth values; it is a contradiction when it is always false.

Two compound statements are equivalent when they take identical truth values for every possible assignment, so one may replace the other in reasoning.

A rule that combines any two elements of a set to produce another element of the same set, such as addition on the integers.

The conditions a set with a binary operation must satisfy to be a group: closure, associativity, an identity element, and an inverse for every element.

A square table displaying the results of a binary operation on every ordered pair of elements of a finite set, useful for spotting identity and inverses.