#### Topics

##### Rational and Irrational Numbers

##### Parallel Lines and Transversal

- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
- Properties of Parallel Lines
- Corresponding Angle Theorem
- Alternate Angles Theorems
- Interior Angle Theorem
- To Draw a Line Parallel to the Given Line Through a Point Outside the Given Line Using Set-square.
- To Draw a Parallel Line to a Given Line at a Given Distance.

##### Indices and Cube Root

- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Meaning of Numbers with Rational Indices
- Concept of Cube Number
- Concept of Cube Root
- Cube Root Through Prime Factorisation Method

##### Altitudes and Medians of a Triangle

##### Expansion Formulae

##### Factorisation of Algebraic Expressions

##### Variation

- Direct Variation
- Inverse Variation
- Time, Work, Speed

##### Quadrilateral : Constructions and Types

- Constructing a Quadrilateral
- Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
- Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
- Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
- Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
- Properties of Rectangle
- Properties of a Square
- Properties of Rhombus
- Properties of a Parallelogram
- Properties of Trapezium
- Properties of Kite

##### Discount and Commission

- Concept of Discount
- Commission
- Rebate

##### Division of Polynomials

- Polynomials
- Degree of Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Divide a Polynomial by a Binomial

##### Statistics

- Arithmetic Mean - Raw Data
- Subdivided Bar Graph
- Percentage Bar Graph

##### Equations in One Variable

- Solution of Equations in One Variable
- Word Problems of Equation in One Variable

##### Congruence of Triangles

- Congruence of Triangles
- Criteria for Congruence of Triangles
- SAS Congruence Criterion
- SSS Congruence Criterion
- ASA Congruence Criterion
- AAS (Or SAA) Test
- RHS Congruence Criterion

##### Compound Interest

##### Area

- Area of a Parallelogram
- Area of a Rhombus
- Area of Trapezium
- Area of a Triangle
- Area of a Triangle by Heron's Formula
- Area of Figures Having Irregular Shape
- Circumference of a Circle
- Area of Circle

##### Surface Area and Volume

##### Circle - Chord and Arc

- Properties of Chord of a Circle
- Arcs Corresponding to the Chord of a Circle

## Formula

- a
^{3 }+ b^{3}= ( a + b )(a^{2}- ab + b^{2})

## Notes

**Factorisation using Identity a**^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}):

^{3}+ b

^{3}= (a + b)(a

^{2}- ab + b

^{2}):

We know that, (a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}, which we can write as

(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)

Now, a^{3} + b^{3} + 3ab(a + b) = (a + b)^{3} .....(Interchanging the sides)

∴ a^{3} + b^{3} = (a + b)^{3} - 3ab(a + b)

= [(a + b)(a + b)^{2}] - 3ab(a + b)

= (a + b)[(a + b)^{2} - 3ab]

= (a + b)(a^{2} + 2ab + b^{2} - 3ab)

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

∴ **a ^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}).**

## Example

**Factorise:** x^{3} + 27y^{3}

x^{3} + 27y^{3}

= x^{3} + (3y)^{3}

= (x + 3y)[x^{2} - x(3y) + (3y)^{2}]

= (x + 3y)[x^{2} - 3xy + 9y^{2}]

## Example

**Factorise:** 8p^{3} + 125q^{3}

8p^{3} + 125q^{3 }

= (2p)^{3 }+ (5q)^{3}

= (2p + 5q)[(2p)^{2} - 2p × 5q + (5q)^{2}]

= (2p + 5q)(4p^{2} - 10pq + 25q^{2})

## Example

**Factorise:** `m^3 + 1/(64m^3)`

`m^3 + 1/(64m^3)`

`= m^3 + (1/(4m))^3`

`= (m + 1/(4m))[m^2 - m xx 1/(4m) + (1/(4m))^2]`

`= (m + 1/(4m))(m^2 - 1/4 + 1/(16m^2))`

## Example

**Factorise: **250p^{3} + 432q^{3}

250p^{3} + 432q^{3}

= 2(125p^{3} + 216q^{3})

= 2[(5p)^{3} + (6q)^{3}]

= 2(5p + 6q)(25p^{2} + 30pq + 36q^{2})