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.
Number System | Calculus | Probability | Trigonometry | Geometry & Mensuration | Algebra & Arithmetic | Coordinate Geometry
Coordinate Geometry
A coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates.
Algebra & Arithmetic
Algebra is the study of mathematical symbols and the rules for manipulating these symbols. Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them - addition, subtraction, multiplication and division.
Geometry & Mensuration
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Mensuration is the part of geometry concerned with ascertaining lengths, areas, and volumes.
Trigonometry
Trigonometry is a word derived from three Greek words - Tri meaning Three, Gon meaning Sides and Metron meaning to measure. Trigonometry is the study of relationships between the sides and angles of a triangle.
Probability
The word chance, possible, probably or likely convey some sense of uncertainty about the occurrence of some events. The entire world is filled with uncertainty. You make decisions affected by uncertainty virtually every day.
Calculus
Calculus is that branch of Mathematics that deals with the study of change in the value of a function as the points in the domain change.
Number System
From time immemorial human beings have been trying to have a count of their belongings- goods, ornaments, jewels, animals, trees, sheeps or goats, etc. by using various techniques. One of the greatest inventions in the history of civilization is the creation of numbers.
Fundamental Operations on Rational Numbers
1. Addition of Rational Numbers
(a) Consider the addition of rational numbers with same denominators: p/q and r/q
Rational Numbers
When an integer a is divided by another non-zero integer b. The following cases arise:
Integers
While dealing with natural numbers and whole numbers, it is not always possible to subtract a number from another. For example, (2 – 3), (3 – 7), and (9 – 20) are all not possible in the system of natural numbers and whole numbers. Thus, it needed another extension of numbers which allow such subtractions.
Natural Numbers
The counting numbers 1, 2, 3, ... constitute the system of natural numbers. These are the numbers which are used in day-to-day life.
Numbers
Counting things is easy now. You can count objects in large numbers and represent them through numerals. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers.
Basic Geometry
The term ‘Geometry’ is the English equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’ means Measurement. According to historians, the geometrical ideas shaped up in ancient times, probably due to the need in art, architecture and measurement. These include occasions when the boundaries of cultivated lands had to be marked without giving room for complaints.
Prime Factorisation
When a number is expressed as a product of its factors, the number is said to be factorised.
Tests for Divisibility of Numbers
You can find a pattern that can tell whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11.
Even and Odd Numbers
In the numbers 2, 4, 6, 8, 10, 12, 14, ... each of them is a multiple of 2. These are called even numbers.
Prime and Composite Numbers
There are numbers, having exactly two factors 1 and the number itself. Such number are 2, 3, 5, 7, 11, and so on. These numbers are prime numbers.
Highest Common Factor (HCF)
You can find the common factors of any two numbers. For example, the common factors of 12 and 16 are 1, 2 and 4. The highest of these common factors is 4.
Lowest Common Multiple (LCM)
The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples. For example, the common multiples of 4 and 6 are 12, 24, 36, ... . The lowest of these is 12. So, the lowest common multiple of 4 and 6 is 12.
Factors and Multiples
A factor of a number is an exact divisor of that number. For example, 1, 2, 3 and 6 are exact divisors of 6. They are called the factors of 6.
Whole Numbers
Given any natural number, you can add 1 to that number and get the next number i.e. you get its successor. The successor of 16 is 16 + 1 = 17, that of 19 is 19 +1 = 20 and so on. The number 16 comes before 17, we say that the predecessor of 17 is 17 – 1 = 16, the predecessor of 20 is 20 – 1 = 19, and so on. Does 1 have both a successor and a predecessor?
Means of Progressions
Arithmetic Mean
A is called the arithmetic mean of the numbers a and b if and only if a, A, b are in AP.
Harmonic Progression
A sequence of non-zero numbers is said to be in harmonic progression (HP) if their reciprocals are in AP.
Geometric Progression
A geometric progression (GP) is a sequence of numbers in which the first term is non-zero and each term, except the first, is obtained by multiplying the term immediately preceding it by a fixed non-zero number. This fixed number is called the common ratio and it is denoted by r.
Arithmetic Progression
An arithmetic progression (AP) is a sequence of numbers in which each term, except the first, is obtained by adding a fixed number to the immediately preceding term. This fixed number is called the common difference, which is generally denoted by d.
Binomial Theorem
A Binomial is an algebraic expression of two terms which are connected by the operation: + or −.
Principle of Mathematical Induction
Induction is the process of inferring a general statement from the truth of particular cases.
Combinations
The word combination means selection. Suppose you are asked to make a selection of any two things from three things: a, b and c.
Circular Permutations
A circular permutation is one in which the things are arranged along a circle. It is also called closed permutation.
Permutations When Objects Can Repeat
The number of permutations of n different things, taken r at a time, when each may be repeated any number of times in each arrangement, is nr.
Permutations of Objects Not All Distinct
The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p + q = n, is n!/p! q!.
Scalar or Dot Product
The scalar product of two vectors is a scalar quantity. Therefore, the product is called scalar product.
Direction Ratios
Any three numbers proportional to direction cosines of a vector are called its direction ratios.
Direction Cosines
Let P(x, y, z) be any point in space with reference to a rectangular coordinate system O (XYZ). Let α, β and γ be the angles made by OP with the positive direction of coordinate axes OX, OY, OZ respectively. Then cos α, cos β, cos γ are called the direction cosines.
Resolution of Vector
If P is a point in a two dimensional plane which has coordinates (x, y).
Addition of Vectors
Triangle Law of Addition of Vectors
Triangle Law of Addition of vectors states that, if two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order.
Types of Vectors
Zero or Null Vector
A vector whose initial and terminal points are coincident is called a zero or null or a void vector. The zero vector is denoted by 
.
Introduction To Vectors
Physical quantities are divided into two categories: Scalar quantities and Vector quantities.
Inverse Trigonometrical Functions
The quantities sin−1x, cos−1x, tan−1x are called inverse circular functions. sin−1x is an angle θ, whose sine is x.
Solution of Triangles
A triangle has six parts or elements - the sides a, b, c and the angles A, B, C.
General Solutions of sin θ = sin α, cos θ = cos α, tan θ = tan α
sin θ = sin α
θ = nπ + (−1)nα, where n ∈ Z