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.

Intervals

The real numbers can be represented geometrically as points on a number line called the real line.

Concept of Permutations

The word permutation means arrangement. For example, given 3 letters a, b, c suppose you arrange them taking 2 at a time. The various arrangements are: ab, ba, bc, cb, ac, ca.

Fundamental Principles of Counting

There are two fundamental principles - principle of addition and principle of multiplication. These two principles enable to understand permutations and combinations and form the base for permutations and combinations.

Factorial

The continued product of first n natural numbers is called the n factorial and is denoted by n!.

Partial Fractions

A given proper fraction can be expressed as the sum of other simple fractions corresponding to the factors of the denominator of the given proper fraction. This process is called splitting into Partial Fractions.

Rational Expressions

An algebraic expression, which can be expressed in the from P/Q, where P and Q (non-zero polynomials) are polynomials, is called a rational expression.

LCM of Polynomials

The Lowest Common Multiple (LCM) of two or more polynomials is the product of the polynomials of the lowest degree and the smallest numerical coefficient which are multiples of the corresponding elements of each of the given polynomials.

HCF of Polynomials

The Highest Common Factor (HCF) of two or more given polynomials is the product of the polynomials of highest degree and greatest numerical coefficient each of which is a factor of each of the given polynomials.

Factorization of Polynomials

Since (x + y)(x – y) = x2 – y2, you can say that (x + y) and (x – y) are factors of the product (x2 – y2).

Algebraic Formulas (Special Products)

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

Product of Determinants

Rule for multiplication of two determinants is the same as the rule for multiplication of two matrices.

Properties of Determinants

Property 1

The value of a determinant is unaltered by interchanging its rows and columns.

Singular and Non-singular Matrices

Singular Matrix

A square matrix A is said to be singular if |A| = 0.

Minors and Co-factors

To evaluate the determinant of order 3 or above, you need to define minors and co-factors.

Introduction to Determinants

The term determinant was first introduced by Gauss in 1801 while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic forms.

Algebraic Properties of Matrices

If A and B are any two matrices of the same order, then

Matrix Multiplication

Two matrices A and B are said to be conformable for multiplication if the number of columns of the first matrix A is equal to the number of rows of the second matrix B.

Two matrices A and B can be added provided both the matrices are of the same order and their sum A + B is obtained by adding the corresponding entries of both the matrices A and B.

Negative of Matrix

Let A be any matrix. The negative of a matrix A is –A and is obtained by changing the sign of all the entries of matrix A.

Scalar Multiplication of Matrix

Let A be any matrix. Let k be any non-zero scalar. The matrix kA is obtained by multiplying all the entries of matrix A by the non-zero scalar k.

Transpose of Matrix

The matrix obtained from the given matrix A by interchanging its rows into columns and its columns into rows is called the transpose of A and it is denoted by A′ or AT.

Equality of Matrices

Two matrices A and B are said to be equal if

Row Matrix

A matrix having only one row is called a row matrix or a row vector.

Order or Size of Matrix

The order or size of a matrix is the number of rows and the number of columns that are present in a matrix.

Introduction to Matrix

The term matrix was first introduced by Sylvester in 1850. He defined a matrix to be an arrangement of terms. In 1858, Cayley outlined a matrix algebra defining addition, multiplication, scalar multiplication and inverses.

Compound Interest

When the interest is calculated on the Principal for the entire period of loan, the interest is called simple interest and is given by

Simple Interest

When a person has to borrow some money as a loan from his friends. relatives or bank, he promises to return it after a specified time period along with some extra money for using the money of the lender.

Discount

A discount is a reduction in the marked (or list) price of an article. For example, 20% discount means a reduction of 20% in the marked price of an article. If the marked price of an article is Rs.100, it is sold for Rs.80, i.e. Rs.20 less than the marked price.

Profit and Loss

Cost Price (CP)

The Price at which an article is purchased, is called its cost price.

Percent

A fraction whose denominator is 100 is read as percent, for example 23/100 is read as twenty three percent or 23%.

Division of Polynomials

The steps involved in the process of division of a polynomial by another polynomial are:

Degree of Polynomial

The sum of the exponents of the variables in a term is called the degree of that term.

Algebraic Expressions

Expressions, involving arithmetical numbers, variables and symbols of operations are called algebraic expressions. An algebraic expression is a combination of numbers, variables and arithmetical operations.

Introduction To Algebra

Sometimes, letters called literal numbers, are used as symbols to represent numbers. For example, when you want to say theta the cost of one book is twenty rupees.

Angle of Elevation and Depression

Angle of Elevation

When the observer is looking at an object (P) which is at a greater height than the observer (A), he has to lift his eyes to see the object and an angle of elevation is formed between the line of sight joining the observer’s eye to the object and the horizontal line.

T-Ratios for Angle of 60°

In ΔPMO and ΔPMM′,

PM = PM (Common)

∠PMO = ∠PMM′ (Each = 90°)

T-Ratios for Angle of 30°

In ΔPMO and ΔP′MO,

OM = OM (Common)

∠PMO = ∠P′MO (Each = 90°)

T-Ratios for Angle of 45°

∠POM + ∠OPM + ∠PMO = 180°

45° + ∠OPM + 90° = 180°

∠OPM = 180° – 90° – 45° = 45°

In ΔPMO, ∠OPM = ∠POM = 45°

T-Ratios of Complementary Angles

Two angles are complementary if their sum is 90°. If the sum of two angles A and B is 90°, then ∠A and ∠B are complementary angles and each of them is complement of the other.

Trigonometric Identities

When equation involving a variable is true for all values of the variable, it is called an identity.

Trigonometric Ratios

Let there be a right triangle ABC, right angled at B. Here ∠A (∠CAB) is an acute angle, AC is hypotenuse, side BC is opposite to ∠A and side AB is adjacent to ∠A.

Secants and Tangents

In case of intersection of a line and a circle, the following three possibilities are there:

Concyclic Points

Points which lie on a circle are called concyclic points.

Circle and Related Terms

A circle is a collection of all points in a plane which are at a constant distance from a fixed point in the same plane.

Properties of Circles

There is one and only one circle passing through three non-collinear points.

Angle Properties

Angle Sum Property of a Triangle

The sum of the three angles of triangle is 180°.

Similarity of Triangles

Any two polygons, with corresponding angles equal and corresponding sides proportional, are similar. Thus, two polygons are similar, if they satisfy the following two conditions:

Properties of Parallelograms

Diagonal & Area of Parallelogram

A diagonal of a parallelogram divides it into two triangles of equal area.

Transversal

A line which intersects two or more lines at distinct points is called a transversal. When a transversal intersects two lines, eight angles are formed.