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

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

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.

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.

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

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**.

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**.

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.

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.

Since (x + y)(x – y) = x^{2} – y^{2}, you can say that (x + y) and (x – y) are factors of the product (x^{2} – y^{2}).

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a – b)^{2} = a^{2} – 2ab + b^{2}

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

**Property 1**

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

**Singular Matrix**

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

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

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.

**1. Matrix addition is commutative.**

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

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.

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.

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.

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 A^{T}.

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

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

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.

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.

**Cost Price (CP)**

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

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

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

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

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.

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**

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.

∠POM + ∠OPM + ∠PMO = 180°

45° + ∠OPM + 90° = 180°

∠OPM = 180° – 90° – 45° = 45°

In ΔPMO, ∠OPM = ∠POM = 45°

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.

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

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.

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

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.

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

**Angle Sum Property of a Triangle**

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

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:

**Diagonal & Area of Parallelogram**

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

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.

Page 4 of 5