Maths

Fundamental Operations on Rational Numbers

Four fundamental operations on rational expressions are performed in exactly the same way as in case of rational 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.

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.

Properties of Combinations

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 n^r.

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!.

Vector or Cross Product

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.

Properties of Principal Inverse Trigonometric Functions

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.

Triangle Area Formula

In any triangle ABC, area of the triangle is

Sub-multiple Half Angle Formula

In any triangle ABC,

Projection Formula

In any triangle ABC,

Cosine Formula

In any triangle ABC,

Napier’s Formula

In any triangle ABC,

Sine Formula

In any triangle ABC,

a/sinA = b/sin B = c/sinC = 2R

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

Equation of the Form: a cosθ + b sinθ = c

a cosθ + b sinθ = c where c² ≤ a² + b² 

Divide each term by √(a² + b²)

General Solutions of sin θ = 0, cos θ = 0, tan θ = 0

sin θ = 0

The general solution of sin θ = 0 is given by

θ = nπ, n ∈ Z where Z is the set of all integers.

Trigonometrical Equations

An equation involving trigonometrical function is called a trigonometrical equation. To solve these equations, you are required to find all replacements for the variable θ that make the equations true.

Trigonometric Conditional Identities

If A + B + C = π, then

Transformation of Product Into Sum or Difference

Trigonometric Multiple Angle Identities

Identities involving sin 2A, cos 2A, tan 3A, etc. are called multiple angle identities. To derive these identities you can use sum identities.

Trigonometric Compound Angles: A + B and A − B

The relation f(x+ y) = f(x) + f(y) is not true for all functions of a real variable. All the six trigonometric ratios do not satisfy this relation.

Properties of Trigonometric Functions

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.

T-Ratios of Related Angles

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.

Quadrants, Angles & Measurement

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 n^th 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^iθ 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