# Maths

Mathematics is the base of human civilization. From cutting vegetables to arranging books on the shelves, from tailoring clothes to motion of Planets - Mathematics applies everywhere.

### Periodic Functions

A function f(x) is said to be a periodic function with period α if f(x + α) = f(x). The least positive value of α is called the fundamental period of the function.

##### Six Trigonometrical Ratios

In the co-ordinate plane, consider a point A on the positive side of x-axis. Let this point revolve about the origin in the anti-clockwise direction through an angle θ and reach the point P.

### Angle and Measurement

An angle is defined as the amount of rotation of a revolving line from the initial position to the terminal position. Counter-clockwise rotations are called positive and the clockwise are called negative.

##### Roots of Complex Number

A number ω is called an nth root of a complex number z, if ωn = z.

##### De Moivre’s Theorem

For any rational number n, cos nθ + i sin nθ is the value or one of the values of (cos θ + i sin θ)n.

##### Cube Roots of Unity

Let x be the cube root of unity, then

x = (1)1/3

x3 = 1

##### Exponential Form of Complex Number

The symbol e or exp (iθ) (called exponential of iθ) is defined by

##### Polar Form of Complex Number

Let (r, θ) be the polar co-ordinates of the point.

P = P(x, y) in the complex plane corresponding to the complex number z = x + iy.

##### Geometrical Representation of Complex Number

If real scales are chosen on two mutually perpendicular axes X′OX and Y′OY (called the x axis and y axis respectively), you can locate any point in the plane determined by these lines, by the ordered pair of real numbers (a, b) called rectangular co-ordinates of the point.

##### Modulus of Complex Number

Let z = a + ib be a complex number. The modulus or absolute value of z denoted by |z| is defined by

##### Conjugate of Complex Number

If z = a + ib, then the conjugate of z is denoted by and is defined by

### Equality of Complex Numbers

Two complex numbers a + ib and c + id are equal if and only if a = c and b = d. The corresponding real parts are equal and the corresponding imaginary parts are equal.

##### Complex Number System

The number system is the gradual development from natural numbers to integers, from integers to rational numbers and from rational numbers to the real numbers.

##### Solution of System of Linear Equations by Matrix Inversion Method

AX = B

If the coefficients of matrix A is non-singular, then A−1 exists.

##### Inverse of a Matrix

Let A be a square matrix of order n. Then a matrix B, if it exists, such that AB = BA = In is called inverse of the matrix A. In this case, A is an invertible matrix.

##### Rank of a Matrix

With each matrix, you can associate a non-negative integer, called its rank. The concept of rank plays an important role in solving a system of homogeneous and non-homogeneous equations.

The concept of division is not defined for matrices. In its place, the notion of the inverse of a matrix is introduced. To define the inverse of a matrix, you need the concept of adjoint of a matrix.

##### Normal Distribution

The Binomial and the Poisson distribution are the most useful theoretical distribution for discrete variables. They relate to the occurrence of distinct events. In order to have mathematical distribution suitable for dealing with quantities whose magnitude is continuously varying, a continuous distribution is needed. The normal distribution is also called the normal probability distribution, is the most useful theoretical distribution for continuous variables.

##### Poisson Distribution

It is named after the French Mathematician Simeon Denis Poisson (1781 −1840) who discovered it. Poisson distribution is a discrete distribution. Poisson distribution is a limiting case of Binomial distribution under the following conditions:

##### Binomial Distribution

This was discovered by a Swiss Mathematician James Bernoulli (1654-1705).

##### Theoretical Distributions

The values of random variables may be distributed according to some definite probability law which can be expressed mathematically and the corresponding probability distribution is called theoretical distribution. Theoretical distributions are based on expectations on the basis of previous experience.

##### Mathematical Expectation

If X denotes a discrete random variable which can assume the values x1, x2, ..., xn with respective probabilities p1, p2, ..., pn then the mathematical expectation of X, denoted by E(X) is defined by

##### Random Variable

The outcomes of an experiment are represented by a random variable if these outcomes are numerical or if real numbers can be assigned to them.

##### Linear Differential Equation

A first order differential equation is said to be linear in y if the power of the terms dy/dx and y are unity.

##### Homogeneous Differential Equation

A differential equation of first order and first degree is said to be homogeneous if it can be put in the form:

##### Variable Separable Method

Variables of a differential equation are to be rearranged in the form:

##### Formation of Differential Equations

Let f (x, y, c1) = 0 be an equation containing x, y and one arbitrary constant c1. If c1 is eliminated by differentiating f (x, y, c1) = 0 with respect to the independent variable once, you get a relation involving x, y and dy/dx, which is a differential equation of the first order.

##### Order and Degree of Differential Equation

The order of a differential equation is the order of the highest order derivative occurring in it. The degree of the differential equation is the degree of the highest order derivative which occurs in it, after the differential equation has been made free from radicals and fractions as far as the derivatives are concerned.

##### Introduction to Differential Equations

An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a Differential Equation.

##### Area of Bounded Regions

Let y = f(x) be a continuous function defined on [a, b], which is positive (f(x) lies on or above x-axis) on the interval [a, b]. Then, the area bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b is given by

##### Reduction Formulae For Integration

A formula which expresses (or reduces) the integral of the nth indexed function in terms of that of (n−1)th indexed (or lower indexed) function is called a reduction formula.

##### Curve Tracing

The study of calculus and its applications is best understood when it is studied through the geometrical representation of the functions involved. In order to investigate the nature of a function (graph) it is not possible to locate each and every point of the graph. But you can sketch the graph of the function and know its nature by certain specific properties and some special points.

##### Differentials: Errors and Approximation

Let y = f(x) be a differentiable function. Then the quantities dx and dy are called differentials. The differential dx is an independent variable that is dx can be given any real number as the value. The differential dy is then defined in terms of dx by the relation

##### Concavity, Convexity and Points of Inflection

If the graph of flies above all of its tangents on an interval I, then it is called concave upward (convex downward) on I. If the graph of flies below all of its tangents on I, it is called concave downward (convex upward) on I.

##### Maximum and Minimum Values

Some of the most important applications of differential calculus are optimization problems, in which you are required to find the optimal (best) way of doing something. These problems can be reduced to finding the maximum or minimum values of a function. Many practical problems require to minimize a cost or maximize an area.

##### Increasing and Decreasing Functions

In sketching the graph of a function, it is very useful to know where it raises and where it falls. The graph shown below raises from A to B, falls from B to C, and raises again from C to D.

##### Indeterminate Forms: L'Hospital's Rule

Suppose f(x) and g(x) are defined on some interval [a,b], such that f(a) = 0 and g(a) = 0, then the ratio f(x)/g(x) is not defined for x = a and gives a meaningless expression 0/0 but has a very definite meaning for values of x ≠ a.

##### Lagrange Mean Value Theorem (LMVT)

Let f(x) be a real valued function that satisfies the following conditions:

##### Rolle’s Theorem

Let f be a real valued function that satisfies the following three conditions:

##### Tangents and Normals: Derivative As Measure Of Slope

Consider a curve whose equation is y = f(x). On this curve, take a point P(x1,y1).

##### Derivative As Rate Measure

If a quantity y depends on and varies with a quantity x, then the rate of change of y with respect to x is dy/dx.

##### Rectangular Hyperbola

A hyperbola is said to be a rectangular hyperbola if its asymptotes are at right angles.

##### Asymptotes

An asymptote to a curve is the tangent to the curve such that the point of contact is at infinity. the asymptote touches the curve at +∞ and −∞.

##### Other Form of Hyperbola

There is another standard hyperbola in which the transverse axis is along y-axis. If the transverse axis is along y-axis and the conjugate axis is along x-axis, then the equation of the hyperbola is

##### Important Definitions Regarding Hyperbola

Focus

The fixed point is called a focus F1(ae, 0) of the hyperbola.

##### Standard Equation of Hyperbola

The locus of a point whose distance from a fixed point bears a constant ratio, greater than one to its distance from a fixed line is called a hyperbola.

##### Important Definitions Regarding Ellipse

Focus

The fixed point is called focus, denoted as F1 (ae, 0).