#### Topics

##### Number Systems

##### Real Numbers

##### Algebra

##### Pair of Linear Equations in Two Variables

- Linear Equations in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Substitution Method
- Elimination Method
- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient

##### Arithmetic Progressions

##### Quadratic Equations

- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Quadratic Equations Examples and Solutions

##### Polynomials

##### Geometry

##### Circles

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

##### Triangles

- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem Or Thales Theorem
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Pythagoras Theorem
- Similarity Triangle Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity
- Ratio of Sides of Triangle

##### Constructions

##### Trigonometry

##### Heights and Distances

##### Trigonometric Identities

##### Introduction to Trigonometry

##### Statistics and Probability

##### Probability

##### Statistics

##### Coordinate Geometry

##### Lines (In Two-dimensions)

##### Mensuration

##### Areas Related to Circles

##### Surface Areas and Volumes

#### definition

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala.

A lemma is a proven statement used for proving another statement.

#### notes

Euclid's division lemma, as the name suggests, its related to the divisibility of four integers or it's based on division. To understand this division lemma let's look at a real-life example. Let's assume that there are seven pens, take three pens at a time and put it in two boxes. Now there will three pens in each box and one pen will be left out.

Now using basic division we can write it as `7/3`=2 with a remainder of 1. Here, 7 is the dividend, 3 is the divisor, 2 is the quotient and 1 is the remainder.

There is one more way to write this, that is, `7= 3 xx 2 + 1`

We can represent the same equation using variables, let's take 7 as 'a', 3 as 'b', 2 as 'q' and 1 as 'r'. So now it will look like `a = b xx q + r`. This is same as Dividend = divisor × quotient+ remainder. This generalized using variables is called Euclid's division lemma which is nothing but will be like this given positive integers 'a' and 'b', there exists unique integers 'q' and 'r' satisfying a = bq+r, where 0 ≤ r < b, that the remainder 'r' can be 0 but less than the divisor 'b'.

Now going back to the example, the number of pens is 7 which is fixed and 3 pens per box that's also fixed then it is obvious that the number of boxes which you can take and the left out pens they are getting fixed by default that is why the word unique integers is used in this explanation because as there will be only one set of solution.

Let us see the application of Euclid's division lemma. Show that every odd positive integer is of form 4q+1 or 4q+3 where q is some integer. As per Euclid's division lemma a=bq+ r. When compared with the given example, b=4 and 0 ≤ r < 4.

By applying the lemma we get the possible value of 'a' as

a=(4q) or (4q+1) or (4q+2) or (4q+3). As per asked in the example odd positive integer is (4q+1) and (4q+3)

#### notes

Another use of Euclid's division lemma is to find HCF ( Highest Common Factors)

Let's assume two numbers that are 32 and 18. The first step will be 32= 18 × 1 + 14 We will continue this step till we get remainder as 0.

Next 18 will become the dividend and 14 will be the divisor, 18 = 14 × 1 + 4

Similarly, 14 will be dividend and 4 will be the divisor, 14 = 4 × 3 + 2

4 will become dividend and 2 will become divisor, 4 = 2 × 2 + 0

Once we get remainder as 0 the divisor that is 2 here will be the HCF. This Euclid's division lemma using numbers.