#### Topics

##### Number Systems

##### Real Numbers

##### Algebra

##### 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

##### Arithmetic Progressions

##### 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

##### 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 (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

##### 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

- Concept of Surface Area, Volume, and Capacity
- 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

##### Internal Assessment

of Squareroot of 2

## Notes

Irrational numbers are numbers that cannot be written in the form of `p/q`, where p and q are integers and q ≠ 0. Example `sqrt3,sqrt2`, π.

## Theorem

1)Theorem: let p be a prime number. If P divides `a^2` the p divides a, where a is a positive integer.

Proof: If `a^2/p` true then `a/p` is also ture.

a=`(p_1p_2...p_n)` as per fundamental theorem, this means

`a^2`= `(p_1p_2..p_n)^2`

= `p_1^2 p_2^2..p_n^2`

given: p divides `a^2`

that means p must belong to pi ,where i lies between 1 to n

Thus, is a factor of a also.

2)Theorem: `sqrt2` is irrational

Proof: Let's assume `sqrt2` is rational. If `sqrt2` is rational the `sqrt2= a/b`, where a and b are co prime.

`sqrt2= a/b`

b `sqrt2`= a

`2b^2`= `a^2` That means 2 divides `a^2`. And hence 2 is prime number it also divides a.

2×k=`a^2` (k is constant)

2×k= a .......eq1

2`b^2`= 2`k^2`= 4`k^2`

`b^2`= 2`k^2`, Therefore 2 divides `b^2`, and since 2 is prime number it also divides b.

That means 2 is factor of a,b but according to our assumption a and b are co prime which

means they dont have common factors. Here our assumption that `sqrt2` is rational is

proven incorrect, thus sqrt is a irrational number.