#### Topics

##### Number Systems

##### Real Numbers

##### Algebra

##### Polynomials

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

- Linear Equation 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

##### Quadratic Equations

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

##### Arithmetic Progressions

##### Coordinate Geometry

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

##### Constructions

- Division of a Line Segment
- Construction of Tangents to a Circle
- Constructions Examples and Solutions

##### Geometry

##### Triangles

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

##### 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

##### Trigonometry

##### Introduction to Trigonometry

- Trigonometry
- Trigonometry
- Trigonometric Ratios
- Trigonometric Ratios and Its Reciprocal
- Trigonometric Ratios of Some Special Angles
- Trigonometric Ratios of Complementary Angles
- Trigonometric Identities
- Proof of Existence
- Relationships Between the Ratios

##### Trigonometric Identities

##### Some Applications of Trigonometry

##### Mensuration

##### Areas Related to Circles

- Perimeter and Area of a Circle - A Review
- Areas of Sector and Segment of a Circle
- Areas of Combinations of Plane Figures
- Circumference of a Circle
- Area of Circle

##### Surface Areas and Volumes

- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone
- Concept of Surface Area, Volume, and Capacity
- Surface Area and Volume of Different Combination of Solid Figures
- Surface Area and Volume of Three Dimensional Figures

##### Statistics and Probability

##### Statistics

##### Probability

##### Internal Assessment

- Significance of the Fundamental Theorem of Arithmetic

## Notes

Around 300 BC, a philosopher known as Euclid of Alexandria understood that all numbers could be split into these two distinct categories. He realized that any number can be divided down over and over until you reach a group of the smallest equal numbers. And by definition, these smallest numbers are always prime numbers. All numbers are built out of smaller primes.

Take any composite number, ex: 125, and break it down, it will always be left with prime numbers i.e. 5 × 5 × 5. Euclid knew that every number could be expressed using a group of smaller primes. No matter what number you choose, it can always be built with the addition of smaller primes. This is the root of his discovery, known as the fundamental theorem of arithmetic. Every possible number has one and only one prime factorization.

Every composite number can be written as the product of the powers of primes. For example, 4 is a composite number that can be split into 2 × 2. Similarly, 9, 18, 21, 33, and 55 could be split into 3 × 3, 3 × 3 × 2, 7 × 3, 11 × 3, and 11 × 5, respectively.

Let's consider an example 4^{n}, where n is a natural number. Check whether there is any value for which 4^{n} ends with the digit zero. If 4^{n} has to end with the digit zero, then its factors must be 5 × 2. Therefore, 4^{n} = k × 5 where k is a constant.

4^{n} = (2 × 2)^{n} i.e. 2 × 2 × 2 × 2 ×......×n

This does not have 5 in the prime factors. Therefore, 4^{n} doesn't end with 0.

Let us learn how to find LCM and HCF by the prime factorisation method. For example, what are the LCF and HCF of 96 and 404 by prime factorisation?

96 = 2 × 2 × 2 ×2 × 3 Therefore 96 = 25 × 3

404 = 2 × 2 × 101 Hence 404 = 22 × 101

While finding HCF, we will take the lowest common prime factor, whereas to find LCM, the highest prime factors are considered.

HCF = 2^{2} = 4

LCM= 25 × 3 × 101 = 9696