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 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
- a3 + b3 = ( a + b )(a2 - ab + b2)
Notes
Factorisation using Identity a3 + b3 = (a + b)(a2 - ab + b2):
We know that, (a + b)3 = a3 + 3a2b + 3ab2 + b3, which we can write as
(a + b)3 = a3 + b3 + 3ab(a + b)
Now, a3 + b3 + 3ab(a + b) = (a + b)3 .....(Interchanging the sides)
∴ a3 + b3 = (a + b)3 - 3ab(a + b)
= [(a + b)(a + b)2] - 3ab(a + b)
= (a + b)[(a + b)2 - 3ab]
= (a + b)(a2 + 2ab + b2 - 3ab)
= (a + b)(a2 - ab + b2)
∴ a3 + b3 = (a + b)(a2 - ab + b2).
Example
Factorise: x3 + 27y3
x3 + 27y3
= x3 + (3y)3
= (x + 3y)[x2 - x(3y) + (3y)2]
= (x + 3y)[x2 - 3xy + 9y2]
Example
Factorise: 8p3 + 125q3
8p3 + 125q3
= (2p)3 + (5q)3
= (2p + 5q)[(2p)2 - 2p × 5q + (5q)2]
= (2p + 5q)(4p2 - 10pq + 25q2)
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: 250p3 + 432q3
250p3 + 432q3
= 2(125p3 + 216q3)
= 2[(5p)3 + (6q)3]
= 2(5p + 6q)(25p2 + 30pq + 36q2)
